S
S
S
3
+
56
−
13
+
8
/
2
2
+
3
=
5
2
x
=
6
x
=
4
a
x
2
+
b
x
+
c
=
0
a
0
x
=
−
b
±
b
2
−
4
a
c
2
a
x
3
−
4
x
2
+
5
x
−
6
2
x
=
6
x
=
4
2
⋅
4
6
8
10
/
5
=
2
p
q
p
q
p
q
a
x
2
+
b
x
+
c
=
0
a
0
x
=
−
b
±
b
2
−
4
a
c
2
a
a
x
2
+
b
x
+
c
=
0
a
0
x
=
−
b
±
b
2
−
4
a
c
2
a
a
x
2
+
b
x
+
c
=
0
a
0
a
x
2
+
b
x
+
c
=
0
x
2
+
b
a
x
=
−
c
a
x
2
+
b
a
x
+
(
b
2
a
)
2
=
(
b
2
a
)
2
−
c
a
(
x
+
b
2
a
)
2
=
b
2
−
4
a
c
4
a
2
x
+
b
2
a
=
±
b
2
−
4
a
c
2
a
x
=
−
b
±
b
2
−
4
a
c
2
a
r
s
r
=
s
p
q
q
p
x
x
A
X
a
A
a
∈
A
a
A
x
X
=
{
x
1
,
x
2
,
…
,
x
n
}
x
1
,
x
2
,
…
,
x
n
X
=
{
x
:
x
satisfies
P
}
x
X
P
E
E
E
=
{
2
,
4
,
6
,
…
}
or
E
=
{
x
:
x
is an even integer and
x
>
0
}
2
∈
E
E
−
3
∉
E
−
3
E
N
=
{
n
:
n
is a natural number
}
=
{
1
,
2
,
3
,
…
}
;
Z
=
{
n
:
n
is an integer
}
=
{
…
,
−
1
,
0
,
1
,
2
,
…
}
;
Q
=
{
r
:
r
is a rational number
}
=
{
p
/
q
:
p
,
q
∈
Z
where
q
0
}
;
R
=
{
x
:
x
is a real number
}
;
C
=
{
z
:
z
is a complex number
}
A
B
A
⊂
B
B
⊃
A
A
B
A
B
{
4
,
5
,
8
}
⊂
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
N
⊂
Z
⊂
Q
⊂
R
⊂
C
B
A
B
⊂
A
B
A
A
B
A
⊄
B
{
4
,
7
,
9
}
⊄
{
2
,
4
,
5
,
8
,
9
}
A
=
B
A
⊂
B
B
⊂
A
∅
A
∪
B
A
B
A
∪
B
=
{
x
:
x
∈
A
or
x
∈
B
}
;
A
B
A
∩
B
=
{
x
:
x
∈
A
and
x
∈
B
}
A
B
A
B
A
=
{
1
,
3
,
5
}
B
=
{
1
,
2
,
3
,
9
}
A
∪
B
=
{
1
,
2
,
3
,
5
,
9
}
and
A
∩
B
=
{
1
,
3
}
⋃
i
=
1
n
A
i
=
A
1
∪
…
∪
A
n
⋂
i
=
1
n
A
i
=
A
1
∩
…
∩
A
n
A
1
,
…
,
A
n
E
O
E
O
A
B
A
∩
B
=
∅
U
A
⊂
U
A
A
′
A
A
′
=
{
x
:
x
∈
U
and
x
∉
A
}
A
B
A
B
A
∖
B
=
A
∩
B
′
=
{
x
:
x
∈
A
and
x
∉
B
}
R
A
=
{
x
∈
R
:
0
<
x
≤
3
}
and
B
=
{
x
∈
R
:
2
≤
x
<
4
}
A
∩
B
=
{
x
∈
R
:
2
≤
x
≤
3
}
A
∪
B
=
{
x
∈
R
:
0
<
x
<
4
}
A
∖
B
=
{
x
∈
R
:
0
<
x
<
2
}
A
′
=
{
x
∈
R
:
x
≤
0
or
x
>
3
}
A
B
C
A
∪
A
=
A
A
∩
A
=
A
A
∖
A
=
∅
A
∪
∅
=
A
A
∩
∅
=
∅
A
∪
(
B
∪
C
)
=
(
A
∪
B
)
∪
C
A
∩
(
B
∩
C
)
=
(
A
∩
B
)
∩
C
A
∪
B
=
B
∪
A
A
∩
B
=
B
∩
A
A
∪
(
B
∩
C
)
=
(
A
∪
B
)
∩
(
A
∪
C
)
A
∩
(
B
∪
C
)
=
(
A
∩
B
)
∪
(
A
∩
C
)
A
∪
A
=
{
x
:
x
∈
A
or
x
∈
A
}
=
{
x
:
x
∈
A
}
=
A
A
∩
A
=
{
x
:
x
∈
A
and
x
∈
A
}
=
{
x
:
x
∈
A
}
=
A
A
∖
A
=
A
∩
A
′
=
∅
A
B
C
A
∪
(
B
∪
C
)
=
A
∪
{
x
:
x
∈
B
or
x
∈
C
}
=
{
x
:
x
∈
A
or
x
∈
B
,
or
x
∈
C
}
=
{
x
:
x
∈
A
or
x
∈
B
}
∪
C
=
(
A
∪
B
)
∪
C
A
∩
(
B
∩
C
)
=
(
A
∩
B
)
∩
C
A
B
(
A
∪
B
)
′
=
A
′
∩
B
′
(
A
∩
B
)
′
=
A
′
∪
B
′
A
∪
B
=
∅
A
B
(
A
∪
B
)
′
⊂
A
′
∩
B
′
(
A
∪
B
)
′
⊃
A
′
∩
B
′
x
∈
(
A
∪
B
)
′
x
∉
A
∪
B
x
A
B
x
∈
A
′
x
∈
B
′
x
∈
A
′
∩
B
′
(
A
∪
B
)
′
⊂
A
′
∩
B
′
x
∈
A
′
∩
B
′
x
∈
A
′
x
∈
B
′
x
∉
A
x
∉
B
x
∉
A
∪
B
x
∈
(
A
∪
B
)
′
(
A
∪
B
)
′
⊃
A
′
∩
B
′
(
A
∪
B
)
′
=
A
′
∩
B
′
(
A
∖
B
)
∩
(
B
∖
A
)
=
∅
(
A
∖
B
)
∩
(
B
∖
A
)
=
(
A
∩
B
′
)
∩
(
B
∩
A
′
)
=
A
∩
A
′
∩
B
∩
B
′
=
∅
A
B
A
×
B
A
B
A
B
A
×
B
=
{
(
a
,
b
)
:
a
∈
A
and
b
∈
B
}
A
=
{
x
,
y
}
B
=
{
1
,
2
,
3
}
C
=
∅
A
×
B
{
(
x
,
1
)
,
(
x
,
2
)
,
(
x
,
3
)
,
(
y
,
1
)
,
(
y
,
2
)
,
(
y
,
3
)
}
A
×
C
=
∅
n
A
1
×
⋯
×
A
n
=
{
(
a
1
,
…
,
a
n
)
:
a
i
∈
A
i
for
i
=
1
,
…
,
n
}
A
=
A
1
=
A
2
=
⋯
=
A
n
A
n
A
×
⋯
×
A
A
n
A
×
⋯
×
A
n
R
3
A
×
B
f
⊂
A
×
B
A
B
a
∈
A
b
∈
B
(
a
,
b
)
∈
f
A
f
B
f
:
A
→
B
A
→
f
B
(
a
,
b
)
∈
A
×
B
f
(
a
)
=
b
f
:
a
↦
b
A
f
f
(
A
)
=
{
f
(
a
)
:
a
∈
A
}
⊂
B
f
A
=
{
1
,
2
,
3
}
B
=
{
a
,
b
,
c
}
f
g
A
B
f
g
1
∈
A
B
g
(
1
)
=
a
g
(
1
)
=
b
f
:
A
→
B
f
:
R
→
R
f
(
x
)
=
x
3
f
:
x
↦
x
3
f
:
Q
→
Z
f
(
p
/
q
)
=
p
1
/
2
=
2
/
4
f
(
1
/
2
)
=
1
2
f
:
A
→
B
f
B
f
(
A
)
=
B
f
a
∈
A
b
∈
B
f
(
a
)
=
b
f
a
1
a
2
f
(
a
1
)
f
(
a
2
)
f
(
a
1
)
=
f
(
a
2
)
a
1
=
a
2
f
:
Z
→
Q
f
(
n
)
=
n
/
1
f
g
:
Q
→
Z
g
(
p
/
q
)
=
p
p
/
q
g
f
:
A
→
B
g
:
B
→
C
f
g
A
C
(
g
∘
f
)
(
x
)
=
g
(
f
(
x
)
)
f
:
A
→
B
g
:
B
→
C
g
∘
f
:
A
→
C
f
(
x
)
=
x
2
g
(
x
)
=
2
x
+
5
(
f
∘
g
)
(
x
)
=
f
(
g
(
x
)
)
=
(
2
x
+
5
)
2
=
4
x
2
+
20
x
+
25
(
g
∘
f
)
(
x
)
=
g
(
f
(
x
)
)
=
2
x
2
+
5
f
∘
g
g
∘
f
f
∘
g
=
g
∘
f
f
(
x
)
=
x
3
g
(
x
)
=
x
3
(
f
∘
g
)
(
x
)
=
f
(
g
(
x
)
)
=
f
(
x
3
)
=
(
x
3
)
3
=
x
(
g
∘
f
)
(
x
)
=
g
(
f
(
x
)
)
=
g
(
x
3
)
=
x
3
3
=
x
2
×
2
A
=
(
a
b
c
d
)
T
A
:
R
2
→
R
2
T
A
(
x
,
y
)
=
(
a
x
+
b
y
,
c
x
+
d
y
)
(
x
,
y
)
R
2
(
a
b
c
d
)
(
x
y
)
=
(
a
x
+
b
y
c
x
+
d
y
)
R
n
R
m
S
=
{
1
,
2
,
3
}
π
:
S
→
S
π
(
1
)
=
2
,
π
(
2
)
=
1
,
π
(
3
)
=
3
π
(
1
2
3
π
(
1
)
π
(
2
)
π
(
3
)
)
=
(
1
2
3
2
1
3
)
S
π
:
S
→
S
S
f
:
A
→
B
g
:
B
→
C
h
:
C
→
D
(
h
∘
g
)
∘
f
=
h
∘
(
g
∘
f
)
f
g
g
∘
f
f
g
g
∘
f
f
g
g
∘
f
h
∘
(
g
∘
f
)
=
(
h
∘
g
)
∘
f
a
∈
A
(
h
∘
(
g
∘
f
)
)
(
a
)
=
h
(
(
g
∘
f
)
(
a
)
)
=
h
(
g
(
f
(
a
)
)
)
=
(
h
∘
g
)
(
f
(
a
)
)
=
(
(
h
∘
g
)
∘
f
)
(
a
)
f
g
c
∈
C
a
∈
A
(
g
∘
f
)
(
a
)
=
g
(
f
(
a
)
)
=
c
g
b
∈
B
g
(
b
)
=
c
a
∈
A
f
(
a
)
=
b
(
g
∘
f
)
(
a
)
=
g
(
f
(
a
)
)
=
g
(
b
)
=
c
S
i
d
S
i
d
S
i
d
(
s
)
=
s
s
∈
S
g
:
B
→
A
f
:
A
→
B
g
∘
f
=
i
d
A
f
∘
g
=
i
d
B
f
−
1
f
f
f
(
x
)
=
x
3
f
−
1
(
x
)
=
x
3
f
(
x
)
=
ln
x
f
−
1
(
x
)
=
e
x
f
(
f
−
1
(
x
)
)
=
f
(
e
x
)
=
ln
e
x
=
x
f
−
1
(
f
(
x
)
)
=
f
−
1
(
ln
x
)
=
e
ln
x
=
x
A
=
(
3
1
5
2
)
A
R
2
R
2
T
A
(
x
,
y
)
=
(
3
x
+
y
,
5
x
+
2
y
)
T
A
A
T
A
−
1
=
T
A
−
1
A
−
1
=
(
2
−
1
−
5
3
)
;
T
A
−
1
(
x
,
y
)
=
(
2
x
−
y
,
−
5
x
+
3
y
)
T
A
−
1
∘
T
A
(
x
,
y
)
=
T
A
∘
T
A
−
1
(
x
,
y
)
=
(
x
,
y
)
T
B
(
x
,
y
)
=
(
3
x
,
0
)
B
=
(
3
0
0
0
)
T
B
−
1
(
x
,
y
)
=
(
a
x
+
b
y
,
c
x
+
d
y
)
(
x
,
y
)
=
T
B
∘
T
B
−
1
(
x
,
y
)
=
(
3
a
x
+
3
b
y
,
0
)
x
y
y
0
π
=
(
1
2
3
2
3
1
)
S
=
{
1
,
2
,
3
}
π
−
1
=
(
1
2
3
3
1
2
)
π
f
:
A
→
B
g
:
B
→
A
g
∘
f
=
i
d
A
g
(
f
(
a
)
)
=
a
a
1
,
a
2
∈
A
f
(
a
1
)
=
f
(
a
2
)
a
1
=
g
(
f
(
a
1
)
)
=
g
(
f
(
a
2
)
)
=
a
2
f
b
∈
B
f
a
∈
A
f
(
a
)
=
b
f
(
g
(
b
)
)
=
b
g
(
b
)
∈
A
a
=
g
(
b
)
f
b
∈
B
f
a
∈
A
f
(
a
)
=
b
f
a
g
g
(
b
)
=
a
f
X
R
⊂
X
×
X
(
x
,
x
)
∈
R
x
∈
X
(
x
,
y
)
∈
R
(
y
,
x
)
∈
R
(
x
,
y
)
(
y
,
z
)
∈
R
(
x
,
z
)
∈
R
R
X
x
∼
y
(
x
,
y
)
∈
R
=
≡
≅
p
q
r
s
q
s
p
/
q
∼
r
/
s
p
s
=
q
r
∼
p
/
q
∼
r
/
s
r
/
s
∼
t
/
u
q
s
u
p
s
=
q
r
r
u
=
s
t
p
s
u
=
q
r
u
=
q
s
t
s
0
p
u
=
q
t
p
/
q
∼
t
/
u
f
g
R
f
(
x
)
∼
g
(
x
)
f
′
(
x
)
=
g
′
(
x
)
∼
f
(
x
)
∼
g
(
x
)
g
(
x
)
∼
h
(
x
)
f
(
x
)
−
g
(
x
)
=
c
1
g
(
x
)
−
h
(
x
)
=
c
2
c
1
c
2
f
(
x
)
−
h
(
x
)
=
(
f
(
x
)
−
g
(
x
)
)
+
(
g
(
x
)
−
h
(
x
)
)
=
c
1
+
c
2
f
′
(
x
)
−
h
′
(
x
)
=
0
f
(
x
)
∼
h
(
x
)
(
x
1
,
y
1
)
(
x
2
,
y
2
)
R
2
(
x
1
,
y
1
)
∼
(
x
2
,
y
2
)
x
1
2
+
y
1
2
=
x
2
2
+
y
2
2
∼
R
2
A
B
2
×
2
2
×
2
A
∼
B
P
P
A
P
−
1
=
B
A
=
(
1
2
−
1
1
)
and
B
=
(
−
18
33
−
11
20
)
A
∼
B
P
A
P
−
1
=
B
P
=
(
2
5
1
3
)
I
2
×
2
I
=
(
1
0
0
1
)
I
A
I
−
1
=
I
A
I
=
A
A
∼
B
P
P
A
P
−
1
=
B
A
=
P
−
1
B
P
=
P
−
1
B
(
P
−
1
)
−
1
A
∼
B
B
∼
C
P
Q
P
A
P
−
1
=
B
Q
B
Q
−
1
=
C
C
=
Q
B
Q
−
1
=
Q
P
A
P
−
1
Q
−
1
=
(
Q
P
)
A
(
Q
P
)
−
1
P
X
X
1
,
X
2
,
…
X
i
∩
X
j
=
∅
i
j
⋃
k
X
k
=
X
∼
X
x
∈
X
[
x
]
=
{
y
∈
X
:
y
∼
x
}
x
∼
X
X
X
P
=
{
X
i
}
X
X
X
i
∼
X
x
∈
X
x
∈
[
x
]
[
x
]
X
=
⋃
x
∈
X
[
x
]
x
,
y
∈
X
[
x
]
=
[
y
]
[
x
]
∩
[
y
]
=
∅
[
x
]
[
y
]
z
∈
[
x
]
∩
[
y
]
z
∼
x
z
∼
y
x
∼
y
[
x
]
⊂
[
y
]
[
y
]
⊂
[
x
]
[
x
]
=
[
y
]
P
=
{
X
i
}
X
x
y
y
x
x
∼
y
y
∼
x
x
y
y
z
x
z
(
p
,
q
)
(
r
,
s
)
f
(
x
)
g
(
x
)
R
2
(
x
1
,
y
1
)
∼
(
x
2
,
y
2
)
x
1
2
+
y
1
2
=
x
2
2
+
y
2
2
r
s
n
∈
N
r
n
s
n
r
s
n
r
−
s
n
r
−
s
=
n
k
k
∈
Z
r
≡
s
(
mod
n
)
a
b
n
41
≡
17
(
mod
8
)
41
−
17
=
24
8
n
Z
r
r
−
r
=
0
n
r
≡
s
(
mod
n
)
r
−
s
=
−
(
s
−
r
)
n
s
−
r
n
s
≡
r
(
mod
n
)
r
≡
s
(
mod
n
)
s
≡
t
(
mod
n
)
k
l
r
−
s
=
k
n
s
−
t
=
l
n
r
−
t
n
r
−
t
=
r
−
s
+
s
−
t
=
k
n
+
l
n
=
(
k
+
l
)
n
r
−
t
n
3
[
0
]
=
{
…
,
−
3
,
0
,
3
,
6
,
…
}
,
[
1
]
=
{
…
,
−
2
,
1
,
4
,
7
,
…
}
,
[
2
]
=
{
…
,
−
1
,
2
,
5
,
8
,
…
}
[
0
]
∪
[
1
]
∪
[
2
]
=
Z
[
0
]
[
1
]
[
2
]
n
n
A
=
{
x
:
x
∈
N
and
x
is even
}
,
B
=
{
x
:
x
∈
N
and
x
is prime
}
,
C
=
{
x
:
x
∈
N
and
x
is a multiple of
5
}
A
∩
B
B
∩
C
A
∪
B
A
∩
(
B
∪
C
)
A
∩
B
=
{
2
}
B
∩
C
=
{
5
}
A
=
{
a
,
b
,
c
}
B
=
{
1
,
2
,
3
}
C
=
{
x
}
D
=
∅
A
×
B
B
×
A
A
×
B
×
C
A
×
D
A
×
B
=
{
(
a
,
1
)
,
(
a
,
2
)
,
(
a
,
3
)
,
(
b
,
1
)
,
(
b
,
2
)
,
(
b
,
3
)
,
(
c
,
1
)
,
(
c
,
2
)
,
(
c
,
3
)
}
A
×
D
=
∅
A
B
A
×
B
=
B
×
A
A
∪
∅
=
A
A
∩
∅
=
∅
A
∪
B
=
B
∪
A
A
∩
B
=
B
∩
A
A
∪
(
B
∩
C
)
=
(
A
∪
B
)
∩
(
A
∪
C
)
x
∈
A
∪
(
B
∩
C
)
x
∈
A
x
∈
B
∩
C
x
∈
A
∪
B
A
∪
C
x
∈
(
A
∪
B
)
∩
(
A
∪
C
)
A
∪
(
B
∩
C
)
⊂
(
A
∪
B
)
∩
(
A
∪
C
)
x
∈
(
A
∪
B
)
∩
(
A
∪
C
)
x
∈
A
∪
B
A
∪
C
x
∈
A
x
B
C
x
∈
A
∪
(
B
∩
C
)
(
A
∪
B
)
∩
(
A
∪
C
)
⊂
A
∪
(
B
∩
C
)
A
∪
(
B
∩
C
)
=
(
A
∪
B
)
∩
(
A
∪
C
)
A
∩
(
B
∪
C
)
=
(
A
∩
B
)
∪
(
A
∩
C
)
A
⊂
B
A
∩
B
=
A
(
A
∩
B
)
′
=
A
′
∪
B
′
A
∪
B
=
(
A
∩
B
)
∪
(
A
∖
B
)
∪
(
B
∖
A
)
(
A
∩
B
)
∪
(
A
∖
B
)
∪
(
B
∖
A
)
=
(
A
∩
B
)
∪
(
A
∩
B
′
)
∪
(
B
∩
A
′
)
=
[
A
∩
(
B
∪
B
′
)
]
∪
(
B
∩
A
′
)
=
A
∪
(
B
∩
A
′
)
=
(
A
∪
B
)
∩
(
A
∪
A
′
)
=
A
∪
B
(
A
∪
B
)
×
C
=
(
A
×
C
)
∪
(
B
×
C
)
(
A
∩
B
)
∖
B
=
∅
(
A
∪
B
)
∖
B
=
A
∖
B
A
∖
(
B
∪
C
)
=
(
A
∖
B
)
∩
(
A
∖
C
)
A
∖
(
B
∪
C
)
=
A
∩
(
B
∪
C
)
′
=
(
A
∩
A
)
∩
(
B
′
∩
C
′
)
=
(
A
∩
B
′
)
∩
(
A
∩
C
′
)
=
(
A
∖
B
)
∩
(
A
∖
C
)
A
∩
(
B
∖
C
)
=
(
A
∩
B
)
∖
(
A
∩
C
)
(
A
∖
B
)
∪
(
B
∖
A
)
=
(
A
∪
B
)
∖
(
A
∩
B
)
f
:
Q
→
Q
f
f
(
p
/
q
)
=
p
+
1
p
−
2
f
(
p
/
q
)
=
3
p
3
q
f
(
p
/
q
)
=
p
+
q
q
2
f
(
p
/
q
)
=
3
p
2
7
q
2
−
p
q
f
(
2
/
3
)
f
(
1
/
2
)
=
3
/
4
f
(
2
/
4
)
=
3
/
8
f
:
R
→
R
f
(
x
)
=
e
x
f
:
Z
→
Z
f
(
n
)
=
n
2
+
3
f
:
R
→
R
f
(
x
)
=
sin
x
f
:
Z
→
Z
f
(
x
)
=
x
2
f
f
(
R
)
=
{
x
∈
R
:
x
>
0
}
f
f
(
R
)
=
{
x
:
−
1
≤
x
≤
1
}
f
:
A
→
B
g
:
B
→
C
f
−
1
g
−
1
(
g
∘
f
)
−
1
=
f
−
1
∘
g
−
1
f
:
N
→
N
f
:
N
→
N
f
(
n
)
=
n
+
1
R
2
(
x
1
,
y
1
)
∼
(
x
2
,
y
2
)
x
1
2
+
y
1
2
=
x
2
2
+
y
2
2
f
:
A
→
B
g
:
B
→
C
f
g
g
∘
f
g
∘
f
g
g
∘
f
f
g
∘
f
f
g
g
∘
f
g
f
x
,
y
∈
A
g
(
f
(
x
)
)
=
(
g
∘
f
)
(
x
)
=
(
g
∘
f
)
(
y
)
=
g
(
f
(
y
)
)
f
(
x
)
=
f
(
y
)
x
=
y
g
∘
f
c
∈
C
c
=
(
g
∘
f
)
(
x
)
=
g
(
f
(
x
)
)
x
∈
A
f
(
x
)
∈
B
g
f
(
x
)
=
x
+
1
x
−
1
f
f
f
∘
f
−
1
f
−
1
∘
f
f
−
1
(
x
)
=
(
x
+
1
)
/
(
x
−
1
)
f
:
X
→
Y
A
1
,
A
2
⊂
X
B
1
,
B
2
⊂
Y
f
(
A
1
∪
A
2
)
=
f
(
A
1
)
∪
f
(
A
2
)
f
(
A
1
∩
A
2
)
⊂
f
(
A
1
)
∩
f
(
A
2
)
f
−
1
(
B
1
∪
B
2
)
=
f
−
1
(
B
1
)
∪
f
−
1
(
B
2
)
f
−
1
(
B
)
=
{
x
∈
X
:
f
(
x
)
∈
B
}
f
−
1
(
B
1
∩
B
2
)
=
f
−
1
(
B
1
)
∩
f
−
1
(
B
2
)
f
−
1
(
Y
∖
B
1
)
=
X
∖
f
−
1
(
B
1
)
y
∈
f
(
A
1
∪
A
2
)
x
∈
A
1
∪
A
2
f
(
x
)
=
y
y
∈
f
(
A
1
)
f
(
A
2
)
y
∈
f
(
A
1
)
∪
f
(
A
2
)
f
(
A
1
∪
A
2
)
⊂
f
(
A
1
)
∪
f
(
A
2
)
y
∈
f
(
A
1
)
∪
f
(
A
2
)
y
∈
f
(
A
1
)
f
(
A
2
)
x
A
1
A
2
f
(
x
)
=
y
x
∈
A
1
∪
A
2
f
(
x
)
=
y
f
(
A
1
)
∪
f
(
A
2
)
⊂
f
(
A
1
∪
A
2
)
f
(
A
1
∪
A
2
)
=
f
(
A
1
)
∪
f
(
A
2
)
x
∼
y
R
x
≥
y
m
∼
n
Z
m
n
>
0
x
∼
y
R
|
x
−
y
|
≤
4
m
∼
n
Z
m
≡
n
(
mod
6
)
0
∼
R
2
(
a
,
b
)
∼
(
c
,
d
)
a
2
+
b
2
≤
c
2
+
d
2
∼
m
×
n
R
n
R
m
x
∼
y
y
∼
x
x
∼
x
X
=
N
∪
{
2
}
x
∼
y
x
+
y
∈
N
R
2
∖
{
(
0
,
0
)
}
(
x
1
,
y
1
)
∼
(
x
2
,
y
2
)
λ
(
x
1
,
y
1
)
=
(
λ
x
2
,
λ
y
2
)
∼
R
2
∖
(
0
,
0
)
P
(
R
)
300
!
10
66
46
3
−
1
=
2
1
+
2
+
⋯
+
n
=
n
(
n
+
1
)
2
n
n
=
1
2
3
4
n
(
n
+
1
)
n
=
1
1
=
1
(
1
+
1
)
2
n
1
+
2
+
⋯
+
n
+
(
n
+
1
)
=
n
(
n
+
1
)
2
+
n
+
1
=
n
2
+
3
n
+
2
2
=
(
n
+
1
)
[
(
n
+
1
)
+
1
]
2
(
n
+
1
)
S
N
S
S
(
n
)
n
∈
N
S
(
n
0
)
n
0
k
k
≥
n
0
S
(
k
)
S
(
k
+
1
)
S
(
n
)
n
n
0
n
≥
3
2
n
>
n
+
4
8
=
2
3
>
3
+
4
=
7
n
0
=
3
2
k
>
k
+
4
k
≥
3
2
k
+
1
=
2
⋅
2
k
>
2
(
k
+
4
)
2
(
k
+
4
)
=
2
k
+
8
>
k
+
5
=
(
k
+
1
)
+
4
k
n
≥
3
10
n
+
1
+
3
⋅
10
n
+
5
9
n
∈
N
n
=
1
10
1
+
1
+
3
⋅
10
+
5
=
135
=
9
⋅
15
9
10
k
+
1
+
3
⋅
10
k
+
5
9
k
≥
1
10
(
k
+
1
)
+
1
+
3
⋅
10
k
+
1
+
5
=
10
k
+
2
+
3
⋅
10
k
+
1
+
50
−
45
=
10
(
10
k
+
1
+
3
⋅
10
k
+
5
)
−
45
9
(
a
+
b
)
n
=
∑
k
=
0
n
(
n
k
)
a
k
b
n
−
k
a
b
n
∈
N
(
n
k
)
=
n
!
k
!
(
n
−
k
)
!
n
n
!
/
(
k
!
(
n
−
k
)
!
)
(
n
+
1
k
)
=
(
n
k
)
+
(
n
k
−
1
)
(
n
k
)
+
(
n
k
−
1
)
=
n
!
k
!
(
n
−
k
)
!
+
n
!
(
k
−
1
)
!
(
n
−
k
+
1
)
!
=
(
n
+
1
)
!
k
!
(
n
+
1
−
k
)
!
=
(
n
+
1
k
)
n
=
1
n
1
(
a
+
b
)
n
+
1
=
(
a
+
b
)
(
a
+
b
)
n
=
(
a
+
b
)
(
∑
k
=
0
n
(
n
k
)
a
k
b
n
−
k
)
=
∑
k
=
0
n
(
n
k
)
a
k
+
1
b
n
−
k
+
∑
k
=
0
n
(
n
k
)
a
k
b
n
+
1
−
k
=
a
n
+
1
+
∑
k
=
1
n
(
n
k
−
1
)
a
k
b
n
+
1
−
k
+
∑
k
=
1
n
(
n
k
)
a
k
b
n
+
1
−
k
+
b
n
+
1
=
a
n
+
1
+
∑
k
=
1
n
[
(
n
k
−
1
)
+
(
n
k
)
]
a
k
b
n
+
1
−
k
+
b
n
+
1
=
∑
k
=
0
n
+
1
(
n
+
1
k
)
a
k
b
n
+
1
−
k
S
(
n
)
n
∈
N
S
(
n
0
)
n
0
S
(
n
0
)
,
S
(
n
0
+
1
)
,
…
,
S
(
k
)
S
(
k
+
1
)
k
≥
n
0
S
(
n
)
n
≥
n
0
S
Z
S
Z
1
S
=
{
n
∈
N
:
n
≥
1
}
1
∈
S
n
∈
S
0
<
1
n
=
n
+
0
<
n
+
1
1
≤
n
<
n
+
1
n
∈
S
n
+
1
S
S
=
N
N
S
S
S
S
k
1
≤
k
≤
n
S
S
n
+
1
S
S
n
+
1
n
+
1
S
S
S
n
S
n
!
n
n
!
=
1
⋅
2
⋅
3
⋯
(
n
−
1
)
⋅
n
1
!
=
1
n
!
=
n
(
n
−
1
)
!
n
>
1
a
b
b
>
0
q
r
a
=
b
q
+
r
0
≤
r
<
b
q
r
q
′
r
′
q
=
q
′
r
=
r
′
q
r
S
=
{
a
−
b
k
:
k
∈
Z
and
a
−
b
k
≥
0
}
0
∈
S
b
a
q
=
a
/
b
r
=
0
0
∉
S
S
a
>
0
a
−
b
⋅
0
∈
S
a
<
0
a
−
b
(
2
a
)
=
a
(
1
−
2
b
)
∈
S
S
∅
S
r
=
a
−
b
q
a
=
b
q
+
r
r
≥
0
r
<
b
r
>
b
a
−
b
(
q
+
1
)
=
a
−
b
q
−
b
=
r
−
b
>
0
a
−
b
(
q
+
1
)
S
a
−
b
(
q
+
1
)
<
a
−
b
q
r
=
a
−
b
q
S
r
≤
b
0
∉
S
r
b
r
<
b
q
r
r
r
′
q
q
′
a
=
b
q
+
r
,
0
≤
r
<
b
and
a
=
b
q
′
+
r
′
,
0
≤
r
′
<
b
b
q
+
r
=
b
q
′
+
r
′
r
′
≥
r
b
(
q
−
q
′
)
=
r
′
−
r
b
r
′
−
r
0
≤
r
′
−
r
≤
r
′
<
b
r
′
−
r
=
0
r
=
r
′
q
=
q
′
a
b
b
=
a
k
k
a
∣
b
d
a
b
d
∣
a
d
∣
b
a
b
d
d
a
b
d
′
a
b
d
′
∣
d
a
b
a
b
d
=
gcd
(
a
,
b
)
gcd
(
24
,
36
)
=
12
gcd
(
120
,
102
)
=
6
a
b
gcd
(
a
,
b
)
=
1
a
b
r
s
gcd
(
a
,
b
)
=
a
r
+
b
s
a
b
S
=
{
a
m
+
b
n
:
m
,
n
∈
Z
and
a
m
+
b
n
>
0
}
S
S
d
=
a
r
+
b
s
d
=
gcd
(
a
,
b
)
a
=
d
q
+
r
′
0
≤
r
′
<
d
r
′
>
0
r
′
=
a
−
d
q
=
a
−
(
a
r
+
b
s
)
q
=
a
−
a
r
q
−
b
s
q
=
a
(
1
−
r
q
)
+
b
(
−
s
q
)
S
d
S
r
′
=
0
d
a
d
b
d
a
b
d
′
a
b
d
′
∣
d
a
=
d
′
h
b
=
d
′
k
d
=
a
r
+
b
s
=
d
′
h
r
+
d
′
k
s
=
d
′
(
h
r
+
k
s
)
d
′
d
d
a
b
a
b
r
s
a
r
+
b
s
=
1
945
2415
2415
=
945
⋅
2
+
525
945
=
525
⋅
1
+
420
525
=
420
⋅
1
+
105
420
=
105
⋅
4
+
0
105
420
105
525
105
945
105
2415
105
945
2415
d
945
2415
d
105
gcd
(
945
,
2415
)
=
105
r
s
945
r
+
2415
s
=
105
105
=
525
+
(
−
1
)
⋅
420
=
525
+
(
−
1
)
⋅
[
945
+
(
−
1
)
⋅
525
]
=
2
⋅
525
+
(
−
1
)
⋅
945
=
2
⋅
[
2415
+
(
−
2
)
⋅
945
]
+
(
−
1
)
⋅
945
=
2
⋅
2415
+
(
−
5
)
⋅
945
r
=
−
5
s
=
2
r
s
r
=
41
s
=
−
16
gcd
(
a
,
b
)
=
d
r
1
>
r
2
>
⋯
>
r
n
=
d
b
=
a
q
1
+
r
1
a
=
r
1
q
2
+
r
2
r
1
=
r
2
q
3
+
r
3
⋮
r
n
−
2
=
r
n
−
1
q
n
+
r
n
r
n
−
1
=
r
n
q
n
+
1
r
s
a
r
+
b
s
=
d
d
=
r
n
=
r
n
−
2
−
r
n
−
1
q
n
=
r
n
−
2
−
q
n
(
r
n
−
3
−
q
n
−
1
r
n
−
2
)
=
−
q
n
r
n
−
3
+
(
1
+
q
n
q
n
−
1
)
r
n
−
2
⋮
=
r
a
+
s
b
d
a
b
d
a
b
p
p
>
1
p
p
p
1
p
n
>
1
a
b
p
p
∣
a
b
p
∣
a
p
∣
b
p
a
p
∣
b
gcd
(
a
,
p
)
=
1
r
s
a
r
+
p
s
=
1
b
=
b
(
a
r
+
p
s
)
=
(
a
b
)
r
+
p
(
b
s
)
p
a
b
p
b
=
(
a
b
)
r
+
p
(
b
s
)
p
1
,
p
2
,
…
,
p
n
P
=
p
1
p
2
⋯
p
n
+
1
P
p
i
1
≤
i
≤
n
p
i
P
−
p
1
p
2
⋯
p
n
=
1
P
p
p
i
P
n
n
>
1
n
=
p
1
p
2
⋯
p
k
p
1
,
…
,
p
k
n
=
q
1
q
2
⋯
q
l
k
=
l
q
i
p
i
n
n
=
2
n
m
1
≤
m
<
n
n
=
p
1
p
2
⋯
p
k
=
q
1
q
2
⋯
q
l
p
1
≤
p
2
≤
⋯
≤
p
k
q
1
≤
q
2
≤
⋯
≤
q
l
p
1
∣
q
i
i
=
1
,
…
,
l
q
1
∣
p
j
j
=
1
,
…
,
k
p
i
q
i
p
1
=
q
i
q
1
=
p
j
p
1
=
q
1
p
1
≤
p
j
=
q
1
≤
q
i
=
p
1
n
′
=
p
2
⋯
p
k
=
q
2
⋯
q
l
k
=
l
q
i
=
p
i
i
=
1
,
…
,
k
S
S
a
a
a
1
a
a
=
a
1
a
2
1
<
a
1
<
a
1
<
a
2
<
a
a
1
∈
S
a
2
∈
S
a
S
a
1
=
p
1
⋯
p
r
a
2
=
q
1
⋯
q
s
a
=
a
1
a
2
=
p
1
⋯
p
r
q
1
⋯
q
s
a
∉
S
n
f
f
(
n
)
n
2
2
n
+
1
n
2
2
5
+
1
=
4,294,967,297
2
4
=
2
+
2
6
=
3
+
3
8
=
3
+
5
…
4
×
10
18
123456792
84
52
r
s
r
(
84
)
+
s
(
52
)
=
gcd
(
84
,
52
)
1
2
+
2
2
+
⋯
+
n
2
=
n
(
n
+
1
)
(
2
n
+
1
)
6
n
∈
N
S
(
1
)
:
[
1
(
1
+
1
)
(
2
(
1
)
+
1
)
]
/
6
=
1
=
1
2
S
(
k
)
:
1
2
+
2
2
+
⋯
+
k
2
=
[
k
(
k
+
1
)
(
2
k
+
1
)
]
/
6
1
2
+
2
2
+
⋯
+
k
2
+
(
k
+
1
)
2
=
[
k
(
k
+
1
)
(
2
k
+
1
)
]
/
6
+
(
k
+
1
)
2
=
[
(
k
+
1
)
(
(
k
+
1
)
+
1
)
(
2
(
k
+
1
)
+
1
)
]
/
6
S
(
k
+
1
)
S
(
n
)
n
1
3
+
2
3
+
⋯
+
n
3
=
n
2
(
n
+
1
)
2
4
n
∈
N
n
!
>
2
n
n
≥
4
S
(
4
)
:
4
!
=
24
>
16
=
2
4
S
(
k
)
:
k
!
>
2
k
(
k
+
1
)
!
=
k
!
(
k
+
1
)
>
2
k
⋅
2
=
2
k
+
1
S
(
k
+
1
)
S
(
n
)
n
x
+
4
x
+
7
x
+
⋯
+
(
3
n
−
2
)
x
=
n
(
3
n
−
1
)
x
2
n
∈
N
10
n
+
1
+
10
n
+
1
3
n
∈
N
4
⋅
10
2
n
+
9
⋅
10
2
n
−
1
+
5
99
n
∈
N
a
1
a
2
⋯
a
n
n
≤
1
n
∑
k
=
1
n
a
k
f
(
n
)
(
x
)
f
(
n
)
n
f
(
f
g
)
(
n
)
(
x
)
=
∑
k
=
0
n
(
n
k
)
f
(
k
)
(
x
)
g
(
n
−
k
)
(
x
)
1
+
2
+
2
2
+
⋯
+
2
n
=
2
n
+
1
−
1
n
∈
N
1
2
+
1
6
+
⋯
+
1
n
(
n
+
1
)
=
n
n
+
1
n
∈
N
x
(
1
+
x
)
n
−
1
≥
n
x
n
=
0
,
1
,
2
,
…
S
(
0
)
:
(
1
+
x
)
0
−
1
=
0
≥
0
=
0
⋅
x
S
(
k
)
:
(
1
+
x
)
k
−
1
≥
k
x
(
1
+
x
)
k
+
1
−
1
=
(
1
+
x
)
(
1
+
x
)
k
−
1
=
(
1
+
x
)
k
+
x
(
1
+
x
)
k
−
1
≥
k
x
+
x
(
1
+
x
)
k
≥
k
x
+
x
=
(
k
+
1
)
x
S
(
k
+
1
)
S
(
n
)
n
X
X
P
(
X
)
X
X
P
(
{
a
,
b
}
)
=
{
∅
,
{
a
}
,
{
b
}
,
{
a
,
b
}
}
n
n
2
n
S
⊂
N
1
∈
S
n
+
1
∈
S
n
∈
S
S
=
N
a
b
gcd
(
a
,
b
)
r
s
gcd
(
a
,
b
)
=
r
a
+
s
b
14
39
234
165
1739
9923
471
562
23771
19945
−
4357
3754
a
b
r
s
a
r
+
b
s
=
1
a
b
1
,
1
,
2
,
3
,
5
,
8
,
13
,
21
,
…
f
1
=
1
f
2
=
1
f
n
+
2
=
f
n
+
1
+
f
n
n
∈
N
f
n
<
2
n
f
n
+
1
f
n
−
1
=
f
n
2
+
(
−
1
)
n
n
≥
2
f
n
=
[
(
1
+
5
)
n
−
(
1
−
5
)
n
]
/
2
n
5
lim
n
→
∞
f
n
/
f
n
+
1
=
(
5
−
1
)
/
2
f
n
f
n
+
1
f
1
=
1
f
2
=
1
f
n
+
2
=
f
n
+
1
+
f
n
a
b
gcd
(
a
,
b
)
=
1
r
s
a
r
+
b
s
=
1
gcd
(
a
,
s
)
=
gcd
(
r
,
b
)
=
gcd
(
r
,
s
)
=
1
x
,
y
∈
N
x
y
x
y
4
k
4
k
+
1
k
a
,
b
,
r
,
s
a
2
+
b
2
=
r
2
a
2
−
b
2
=
s
2
a
r
s
b
n
∈
N
n
0
,
1
,
…
,
n
−
1
r
s
Z
0
≤
s
<
n
[
r
]
=
[
s
]
n
a
b
lcm
(
a
,
b
)
m
a
b
m
a
b
n
m
n
m
n
a
b
d
=
gcd
(
a
,
b
)
m
=
lcm
(
a
,
b
)
d
m
=
|
a
b
|
lcm
(
a
,
b
)
=
a
b
gcd
(
a
,
b
)
=
1
gcd
(
a
,
c
)
=
gcd
(
b
,
c
)
=
1
gcd
(
a
b
,
c
)
=
1
a
b
c
a
,
b
,
c
∈
Z
gcd
(
a
,
b
)
=
1
a
∣
b
c
a
∣
c
gcd
(
a
,
b
)
=
1
r
s
a
r
+
b
s
=
1
a
c
r
+
b
c
s
=
c
p
≥
2
2
p
−
1
p
6
n
+
5
2
3
6
n
+
1
6
n
+
5
6
k
+
5
4
n
−
1
2
p
q
p
2
=
2
q
2
2
N
n
1
<
n
<
N
2
3
5
4
N
N
N
=
250
N
N
N
=
120
N
N
0
=
N
∪
{
0
}
A
:
N
0
×
N
0
→
N
0
A
(
0
,
y
)
=
y
+
1
,
A
(
x
+
1
,
0
)
=
A
(
x
,
1
)
,
A
(
x
+
1
,
y
+
1
)
=
A
(
x
,
A
(
x
+
1
,
y
)
)
A
(
3
,
1
)
A
(
4
,
1
)
A
(
5
,
1
)
a
b
gcd
(
a
,
b
)
r
s
gcd
(
a
,
b
)
=
r
a
+
s
b
a
b
r
0
≤
r
<
b
a
=
b
q
+
r
q
(
a
−
r
)
/
b
q
b
a
a
b
a
b
a
b
a
b
r
s
r
a
+
s
b
=
gcd
(
a
,
b
)
2
a
a
b
−
1
2600
=
2
3
×
5
2
×
13
2600
1
1
c
=
4
598
037
234
d
=
7
d
=
11
Z
2
×
2
2
×
2
n
n
a
b
n
n
a
−
b
n
Z
n
Z
n
n
12
[
0
]
=
{
…
,
−
12
,
0
,
12
,
24
,
…
}
,
[
1
]
=
{
…
,
−
11
,
1
,
13
,
25
,
…
}
,
⋮
[
11
]
=
{
…
,
−
1
,
11
,
23
,
35
,
…
}
0
,
1
,
…
,
11
[
0
]
,
[
1
]
,
…
,
[
11
]
Z
n
a
b
n
(
a
+
b
)
(
mod
n
)
a
+
b
n
n
(
a
b
)
(
mod
n
)
a
b
n
n
7
+
4
≡
1
(
mod
5
)
7
⋅
3
≡
1
(
mod
5
)
3
+
5
≡
0
(
mod
8
)
3
⋅
5
≡
7
(
mod
8
)
3
+
4
≡
7
(
mod
12
)
3
⋅
4
≡
0
(
mod
12
)
n
0
n
Z
n
Z
8
2
4
6
n
=
2
4
6
k
k
n
≡
1
(
mod
8
)
Z
8
⋅
0
1
2
3
4
5
6
7
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
2
0
2
4
6
0
2
4
6
3
0
3
6
1
4
7
2
5
4
0
4
0
4
0
4
0
4
5
0
5
2
7
4
1
6
3
6
0
6
4
2
0
6
4
2
7
0
7
6
5
4
3
2
1
Z
n
n
a
,
b
,
c
∈
Z
n
a
+
b
≡
b
+
a
(
mod
n
)
a
b
≡
b
a
(
mod
n
)
(
a
+
b
)
+
c
≡
a
+
(
b
+
c
)
(
mod
n
)
(
a
b
)
c
≡
a
(
b
c
)
(
mod
n
)
a
+
0
≡
a
(
mod
n
)
a
⋅
1
≡
a
(
mod
n
)
a
(
b
+
c
)
≡
a
b
+
a
c
(
mod
n
)
a
−
a
a
+
(
−
a
)
≡
0
(
mod
n
)
a
gcd
(
a
,
n
)
=
1
b
a
(
mod
n
)
b
a
b
≡
1
(
mod
n
)
n
a
+
b
n
b
+
a
n
gcd
(
a
,
n
)
=
1
r
s
a
r
+
n
s
=
1
n
s
=
1
−
a
r
a
r
≡
1
(
mod
n
)
b
r
a
b
≡
1
(
mod
n
)
b
a
b
≡
1
(
mod
n
)
n
a
b
−
1
k
a
b
−
n
k
=
1
d
=
gcd
(
a
,
n
)
d
a
b
−
n
k
d
1
d
=
1
180
∘
360
∘
90
∘
△
A
B
C
△
A
B
C
A
B
C
S
π
:
S
→
S
3
!
=
6
3
⋅
2
⋅
1
=
3
!
=
6
A
B
B
C
C
A
(
A
B
C
B
C
A
)
120
∘
△
A
B
C
μ
1
ρ
1
ρ
1
μ
1
(
μ
1
ρ
1
)
(
A
)
=
μ
1
(
ρ
1
(
A
)
)
=
μ
1
(
B
)
=
C
(
μ
1
ρ
1
)
(
B
)
=
μ
1
(
ρ
1
(
B
)
)
=
μ
1
(
C
)
=
B
(
μ
1
ρ
1
)
(
C
)
=
μ
1
(
ρ
1
(
C
)
)
=
μ
1
(
A
)
=
A
μ
2
ρ
1
μ
1
μ
3
ρ
1
μ
1
μ
1
ρ
1
△
A
B
C
α
β
α
β
=
i
d
∘
i
d
ρ
1
ρ
2
μ
1
μ
2
μ
3
i
d
i
d
ρ
1
ρ
2
μ
1
μ
2
μ
3
ρ
1
ρ
1
ρ
2
i
d
μ
3
μ
1
μ
2
ρ
2
ρ
2
i
d
ρ
1
μ
2
μ
3
μ
1
μ
1
μ
1
μ
2
μ
3
i
d
ρ
1
ρ
2
μ
2
μ
2
μ
3
μ
1
ρ
2
i
d
ρ
1
μ
3
μ
3
μ
1
μ
2
ρ
1
ρ
2
i
d
n
G
G
×
G
→
G
(
a
,
b
)
∈
G
×
G
a
∘
b
a
b
G
a
b
(
G
,
∘
)
G
(
a
,
b
)
↦
a
∘
b
(
a
∘
b
)
∘
c
=
a
∘
(
b
∘
c
)
a
,
b
,
c
∈
G
e
∈
G
a
∈
G
e
∘
a
=
a
∘
e
=
a
a
∈
G
a
−
1
a
∘
a
−
1
=
a
−
1
∘
a
=
e
G
a
∘
b
=
b
∘
a
a
,
b
∈
G
Z
=
{
…
,
−
1
,
0
,
1
,
2
,
…
}
m
,
n
∈
Z
+
∘
m
+
n
m
∘
n
0
n
∈
Z
−
n
n
−
1
m
+
n
=
n
+
m
a
b
a
∘
b
m
+
n
−
n
m
−
n
m
+
(
−
n
)
n
n
Z
5
0
1
2
3
4
Z
5
m
+
n
Z
5
2
+
3
=
3
+
2
=
0
Z
5
Z
n
=
{
0
,
1
,
…
,
n
−
1
}
n
(
Z
5
,
+
)
+
0
1
2
3
4
0
0
1
2
3
4
1
1
2
3
4
0
2
2
3
4
0
1
3
3
4
0
1
2
4
4
0
1
2
3
Z
n
Z
n
1
⋅
k
=
k
⋅
1
=
k
k
∈
Z
n
0
0
⋅
k
=
k
⋅
0
=
0
k
Z
n
Z
n
∖
{
0
}
2
∈
Z
6
0
⋅
2
=
0
1
⋅
2
=
2
2
⋅
2
=
4
3
⋅
2
=
0
4
⋅
2
=
2
5
⋅
2
=
4
k
Z
n
k
n
Z
n
U
(
n
)
Z
n
U
(
n
)
Z
n
U
(
8
)
U
(
8
)
⋅
1
3
5
7
1
1
3
5
7
3
3
1
7
5
5
5
7
1
3
7
7
5
3
1
α
β
=
β
α
α
β
S
3
D
3
M
2
(
R
)
2
×
2
G
L
2
(
R
)
M
2
(
R
)
n
×
n
R
A
=
(
a
b
c
d
)
G
L
2
(
R
)
A
−
1
A
A
−
1
=
A
−
1
A
=
I
I
2
×
2
A
A
det
A
=
a
d
−
b
c
0
A
I
=
(
1
0
0
1
)
A
∈
G
L
2
(
R
)
A
−
1
=
1
a
d
−
b
c
(
d
−
b
−
c
a
)
A
B
=
B
A
G
L
2
(
R
)
1
=
(
1
0
0
1
)
I
=
(
0
1
−
1
0
)
J
=
(
0
i
i
0
)
K
=
(
i
0
0
−
i
)
i
2
=
−
1
I
2
=
J
2
=
K
2
=
−
1
I
J
=
K
J
K
=
I
K
I
=
J
J
I
=
−
K
K
J
=
−
I
I
K
=
−
J
Q
8
=
{
±
1
,
±
I
,
±
J
,
±
K
}
Q
8
C
∗
C
∗
1
z
=
a
+
b
i
z
−
1
=
a
−
b
i
a
2
+
b
2
z
G
n
|
G
|
=
n
Z
5
5
Z
|
Z
|
=
∞
G
e
∈
G
e
g
=
g
e
=
g
g
∈
G
e
e
′
G
e
g
=
g
e
=
g
e
′
g
=
g
e
′
=
g
g
∈
G
e
=
e
′
e
e
e
′
=
e
′
e
′
e
e
′
=
e
e
=
e
e
′
=
e
′
g
′
g
″
g
G
g
g
′
=
g
′
g
=
e
g
g
″
=
g
″
g
=
e
g
′
=
g
″
g
′
=
g
′
e
=
g
′
(
g
g
″
)
=
(
g
′
g
)
g
″
=
e
g
″
=
g
″
g
G
g
g
−
1
G
a
,
b
∈
G
(
a
b
)
−
1
=
b
−
1
a
−
1
a
,
b
∈
G
a
b
b
−
1
a
−
1
=
a
e
a
−
1
=
a
a
−
1
=
e
b
−
1
a
−
1
a
b
=
e
(
a
b
)
−
1
=
b
−
1
a
−
1
G
a
∈
G
(
a
−
1
)
−
1
=
a
a
−
1
(
a
−
1
)
−
1
=
e
a
(
a
−
1
)
−
1
=
e
(
a
−
1
)
−
1
=
a
a
−
1
(
a
−
1
)
−
1
=
a
e
=
a
a
b
G
x
∈
G
a
x
=
b
x
G
a
b
G
a
x
=
b
x
a
=
b
G
a
x
=
b
x
a
x
=
b
a
−
1
x
=
e
x
=
a
−
1
a
x
=
a
−
1
b
x
1
x
2
a
x
=
b
a
x
1
=
b
=
a
x
2
x
1
=
a
−
1
a
x
1
=
a
−
1
a
x
2
=
x
2
x
a
=
b
G
a
,
b
,
c
∈
G
b
a
=
c
a
b
=
c
a
b
=
a
c
b
=
c
G
g
∈
G
g
0
=
e
n
∈
N
g
n
=
g
⋅
g
⋯
g
⏟
n
times
g
−
n
=
g
−
1
⋅
g
−
1
⋯
g
−
1
⏟
n
times
g
,
h
∈
G
g
m
g
n
=
g
m
+
n
m
,
n
∈
Z
(
g
m
)
n
=
g
m
n
m
,
n
∈
Z
(
g
h
)
n
=
(
h
−
1
g
−
1
)
−
n
n
∈
Z
G
(
g
h
)
n
=
g
n
h
n
(
g
h
)
n
g
n
h
n
Z
Z
n
n
g
g
n
m
g
+
n
g
=
(
m
+
n
)
g
m
,
n
∈
Z
m
(
n
g
)
=
(
m
n
)
g
m
,
n
∈
Z
m
(
g
+
h
)
=
m
g
+
m
h
n
∈
Z
Z
Z
n
2
Z
=
{
…
,
−
2
,
0
,
2
,
4
,
…
}
H
G
H
G
G
H
H
G
H
=
{
e
}
G
G
R
∗
1
a
∈
R
∗
1
/
a
Q
∗
=
{
p
/
q
:
p
and
q
are nonzero integers
}
R
∗
R
∗
1
1
=
1
/
1
R
∗
Q
∗
Q
∗
p
/
q
r
/
s
p
r
/
q
s
Q
∗
p
/
q
∈
Q
∗
Q
∗
(
p
/
q
)
−
1
=
q
/
p
R
∗
Q
∗
C
∗
H
=
{
1
,
−
1
,
i
,
−
i
}
H
C
∗
H
H
⊂
C
∗
S
L
2
(
R
)
G
L
2
(
R
)
A
=
(
a
b
c
d
)
S
L
2
(
R
)
a
d
−
b
c
=
1
S
L
2
(
R
)
2
×
2
S
L
2
(
R
)
A
A
−
1
=
(
d
−
b
−
c
a
)
S
L
2
(
R
)
H
G
G
H
G
G
2
×
2
M
2
(
R
)
2
×
2
M
2
(
R
)
M
2
(
R
)
(
1
0
0
1
)
+
(
−
1
0
0
−
1
)
=
(
0
0
0
0
)
G
L
2
(
R
)
Z
4
0
2
Z
2
Z
2
×
Z
2
(
a
,
b
)
+
(
c
,
d
)
=
(
a
+
c
,
b
+
d
)
Z
2
×
Z
2
Z
2
×
Z
2
H
1
=
{
(
0
,
0
)
,
(
0
,
1
)
}
H
2
=
{
(
0
,
0
)
,
(
1
,
0
)
}
H
3
=
{
(
0
,
0
)
,
(
1
,
1
)
}
Z
4
Z
2
×
Z
2
Z
2
×
Z
2
+
(
0
,
0
)
(
0
,
1
)
(
1
,
0
)
(
1
,
1
)
(
0
,
0
)
(
0
,
0
)
(
0
,
1
)
(
1
,
0
)
(
1
,
1
)
(
0
,
1
)
(
0
,
1
)
(
0
,
0
)
(
1
,
1
)
(
1
,
0
)
(
1
,
0
)
(
1
,
0
)
(
1
,
1
)
(
0
,
0
)
(
0
,
1
)
(
1
,
1
)
(
1
,
1
)
(
1
,
0
)
(
0
,
1
)
(
0
,
0
)
H
G
e
G
H
h
1
,
h
2
∈
H
h
1
h
2
∈
H
h
∈
H
h
−
1
∈
H
H
G
H
e
H
e
H
=
e
e
G
e
H
e
H
=
e
H
e
e
H
=
e
H
e
=
e
H
e
e
H
=
e
H
e
H
e
=
e
H
H
h
∈
H
H
h
′
∈
H
h
h
′
=
h
′
h
=
e
G
h
′
=
h
−
1
H
G
H
G
H
G
H
∅
g
,
h
∈
H
g
h
−
1
H
H
G
g
h
−
1
∈
H
g
h
H
h
H
h
−
1
H
g
h
−
1
∈
H
H
⊂
G
H
∅
g
h
−
1
∈
H
g
,
h
∈
H
g
∈
H
g
g
−
1
=
e
H
g
∈
H
e
g
−
1
=
g
−
1
H
h
1
,
h
2
∈
H
H
h
1
(
h
2
−
1
)
−
1
=
h
1
h
2
∈
H
H
G
Z
8
6
+
7
2
−
1
U
(
16
)
5
⋅
7
3
−
1
x
∈
Z
3
x
≡
2
(
mod
7
)
5
x
+
1
≡
13
(
mod
23
)
5
x
+
1
≡
13
(
mod
26
)
9
x
≡
3
(
mod
5
)
5
x
≡
1
(
mod
6
)
3
x
≡
1
(
mod
6
)
3
+
7
Z
=
{
…
,
−
4
,
3
,
10
,
…
}
18
+
26
Z
5
+
6
Z
G
=
{
a
,
b
,
c
,
d
}
∘
a
b
c
d
a
a
c
d
a
b
b
b
c
d
c
c
d
a
b
d
d
a
b
c
∘
a
b
c
d
a
a
b
c
d
b
b
a
d
c
c
c
d
a
b
d
d
c
b
a
∘
a
b
c
d
a
a
b
c
d
b
b
c
d
a
c
c
d
a
b
d
d
a
b
c
∘
a
b
c
d
a
a
b
c
d
b
b
a
c
d
c
c
b
a
d
d
d
d
b
c
(
Z
4
,
+
)
D
4
U
(
12
)
⋅
1
5
7
11
1
1
5
7
11
5
5
1
11
7
7
7
11
1
5
11
11
7
5
1
S
=
R
∖
{
−
1
}
S
a
∗
b
=
a
+
b
+
a
b
(
S
,
∗
)
A
B
G
L
2
(
R
)
A
B
B
A
S
L
2
(
R
)
(
1
x
y
0
1
z
0
0
1
)
(
1
x
y
0
1
z
0
0
1
)
(
1
x
′
y
′
0
1
z
′
0
0
1
)
=
(
1
x
+
x
′
y
+
y
′
+
x
z
′
0
1
z
+
z
′
0
0
1
)
det
(
A
B
)
=
det
(
A
)
det
(
B
)
G
L
2
(
R
)
G
L
2
(
R
)
A
B
G
L
2
(
R
)
A
B
∈
G
L
2
(
R
)
Z
2
n
=
{
(
a
1
,
a
2
,
…
,
a
n
)
:
a
i
∈
Z
2
}
Z
2
n
(
a
1
,
a
2
,
…
,
a
n
)
+
(
b
1
,
b
2
,
…
,
b
n
)
=
(
a
1
+
b
1
,
a
2
+
b
2
,
…
,
a
n
+
b
n
)
Z
2
n
R
∗
=
R
∖
{
0
}
R
∗
Z
G
=
R
∗
×
Z
∘
G
(
a
,
m
)
∘
(
b
,
n
)
=
(
a
b
,
m
+
n
)
G
G
g
,
h
∈
G
(
g
h
)
n
g
n
h
n
n
!
n
σ
=
(
1
2
⋯
n
a
1
a
2
⋯
a
n
)
S
n
a
i
n
a
1
n
−
1
a
2
,
…
a
n
−
1
a
n
σ
n
(
n
−
1
)
⋯
2
⋅
1
=
n
!
0
+
a
≡
a
+
0
≡
a
(
mod
n
)
a
∈
Z
n
n
a
⋅
1
≡
a
(
mod
n
)
a
∈
Z
n
b
∈
Z
n
a
+
b
≡
b
+
a
≡
0
(
mod
n
)
n
n
n
n
a
(
b
+
c
)
≡
a
b
+
a
c
(
mod
n
)
a
b
G
a
b
n
a
−
1
=
(
a
b
a
−
1
)
n
n
∈
Z
(
a
b
a
−
1
)
n
=
(
a
b
a
−
1
)
(
a
b
a
−
1
)
⋯
(
a
b
a
−
1
)
=
a
b
(
a
a
−
1
)
b
(
a
a
−
1
)
b
⋯
b
(
a
a
−
1
)
b
a
−
1
=
a
b
n
a
−
1
U
(
n
)
Z
n
n
>
2
k
∈
U
(
n
)
k
2
=
1
k
1
g
1
g
2
⋯
g
n
g
n
−
1
g
n
−
1
−
1
⋯
g
1
−
1
G
a
,
b
∈
G
x
a
=
b
G
G
G
b
a
=
c
a
b
=
c
a
b
=
a
c
b
=
c
a
,
b
,
c
∈
G
a
2
=
e
a
G
G
a
b
a
b
=
(
a
b
)
2
=
e
=
a
2
b
2
=
a
a
b
b
b
a
=
a
b
G
a
∈
G
a
a
2
=
e
G
(
a
b
)
2
=
a
2
b
2
a
b
G
G
Z
3
×
Z
3
Z
3
×
Z
3
Z
9
H
1
=
{
i
d
}
H
2
=
{
i
d
,
ρ
1
,
ρ
2
}
H
3
=
{
i
d
,
μ
1
}
H
4
=
{
i
d
,
μ
2
}
H
5
=
{
i
d
,
μ
3
}
S
3
H
=
{
2
k
:
k
∈
Z
}
H
Q
∗
n
=
0
,
1
,
2
,
…
n
Z
=
{
n
k
:
k
∈
Z
}
n
Z
Z
Z
T
=
{
z
∈
C
∗
:
|
z
|
=
1
}
T
C
∗
G
2
×
2
(
cos
θ
−
sin
θ
sin
θ
cos
θ
)
θ
∈
R
G
S
L
2
(
R
)
G
=
{
a
+
b
2
:
a
,
b
∈
Q
and
a
and
b
are not both zero
}
R
∗
G
1
=
1
+
0
2
(
a
+
b
2
)
(
c
+
d
2
)
=
(
a
c
+
2
b
d
)
+
(
a
d
+
b
c
)
2
G
(
a
+
b
2
)
−
1
=
a
/
(
a
2
−
2
b
2
)
−
b
2
/
(
a
2
−
2
b
2
)
G
2
×
2
H
=
{
(
a
b
c
d
)
:
a
+
d
=
0
}
H
G
S
L
2
(
Z
)
2
×
2
S
L
2
(
R
)
Q
8
G
G
H
K
G
H
∪
K
G
S
3
H
K
G
H
K
=
{
h
k
:
h
∈
H
and
k
∈
K
}
G
G
G
g
∈
G
Z
(
G
)
=
{
x
∈
G
:
g
x
=
x
g
for all
g
∈
G
}
G
G
a
b
G
a
4
b
=
b
a
a
3
=
e
a
b
=
b
a
b
a
=
a
4
b
=
a
3
a
b
=
a
b
x
y
=
x
−
1
y
−
1
x
y
G
G
H
G
C
(
H
)
=
{
g
∈
G
:
g
h
=
h
g
for all
h
∈
H
}
C
(
H
)
G
H
G
H
G
g
∈
G
g
H
g
−
1
=
{
g
h
g
−
1
:
h
∈
H
}
G
d
1
d
2
⋯
d
12
3
⋅
d
1
+
1
⋅
d
2
+
3
⋅
d
3
+
⋯
+
3
⋅
d
11
+
1
⋅
d
12
≡
0
(
mod
10
)
d
12
(
d
1
,
d
2
,
…
,
d
k
)
⋅
(
w
1
,
w
2
,
…
,
w
k
)
≡
0
(
mod
n
)
d
1
w
1
+
d
2
w
2
+
⋯
+
d
k
w
k
≡
0
(
mod
n
)
(
d
1
,
d
2
,
…
,
d
k
)
⋅
(
w
1
,
w
2
,
…
,
w
k
)
≡
0
(
mod
n
)
k
d
1
d
2
⋯
d
k
0
≤
d
i
<
n
gcd
(
w
i
,
n
)
=
1
1
≤
i
≤
k
(
d
1
,
d
2
,
…
,
d
k
)
⋅
(
w
1
,
w
2
,
…
,
w
k
)
≡
0
(
mod
n
)
k
d
1
d
2
⋯
d
k
0
≤
d
i
<
n
d
i
d
j
gcd
(
w
i
−
w
j
,
n
)
=
1
i
j
1
k
(
d
1
,
d
2
,
…
,
d
10
)
⋅
(
10
,
9
,
…
,
1
)
≡
0
(
mod
11
)
d
10
6
6
1
6.00000
6.00000
+
0.00000
i
6
8
n
n
ρ
2
ρ
2
=
(
A
B
C
C
A
B
)
=
(
1
2
3
3
1
2
)
f
g
(
f
g
)
(
x
)
=
f
(
g
(
x
)
)
g
f
g
f
…
μ
ρ
−
1
I
J
K
1
−
1
−
1
⋅
−
1
=
1
I
−
I
i
=
−
1
S
8
4
4
Z
Z
n
3
∈
Z
3
3
Z
=
{
…
,
−
3
,
0
,
3
,
6
,
…
}
3
Z
3
3
3
H
=
{
2
n
:
n
∈
Z
}
H
Q
∗
a
=
2
m
b
=
2
n
H
a
b
−
1
=
2
m
2
−
n
=
2
m
−
n
H
H
Q
∗
2
G
a
G
a
⟨
a
⟩
=
{
a
k
:
k
∈
Z
}
G
⟨
a
⟩
G
a
⟨
a
⟩
a
0
=
e
g
h
⟨
a
⟩
⟨
a
⟩
g
=
a
m
h
=
a
n
m
n
g
h
=
a
m
a
n
=
a
m
+
n
⟨
a
⟩
g
=
a
n
⟨
a
⟩
g
−
1
=
a
−
n
⟨
a
⟩
H
G
a
a
H
⟨
a
⟩
⟨
a
⟩
G
a
⟨
a
⟩
=
{
n
a
:
n
∈
Z
}
a
∈
G
⟨
a
⟩
a
G
a
G
=
⟨
a
⟩
G
a
G
a
G
a
n
a
n
=
e
|
a
|
=
n
a
n
a
|
a
|
=
∞
a
1
5
Z
6
Z
6
2
∈
Z
6
3
2
⟨
2
⟩
=
{
0
,
2
,
4
}
Z
Z
n
1
−
1
Z
Z
n
Z
n
Z
6
U
(
9
)
Z
9
U
(
9
)
{
1
,
2
,
4
,
5
,
7
,
8
}
U
(
9
)
2
1
=
2
2
2
=
4
2
3
=
8
2
4
=
7
2
5
=
5
2
6
=
1
S
3
S
3
S
3
G
a
∈
G
G
g
h
G
a
g
=
a
r
h
=
a
s
g
h
=
a
r
a
s
=
a
r
+
s
=
a
s
+
r
=
a
s
a
r
=
h
g
G
G
G
G
G
G
a
H
G
H
=
{
e
}
H
H
g
g
a
n
n
H
g
−
1
=
a
−
n
H
n
−
n
H
a
n
>
0
m
a
m
∈
H
m
h
=
a
m
H
h
′
∈
H
h
h
′
∈
H
H
G
h
′
=
a
k
k
q
r
k
=
m
q
+
r
0
≤
r
<
m
a
k
=
a
m
q
+
r
=
(
a
m
)
q
a
r
=
h
q
a
r
a
r
=
a
k
h
−
q
a
k
h
−
q
H
a
r
H
m
a
m
H
r
=
0
k
=
m
q
h
′
=
a
k
=
a
m
q
=
h
q
H
h
Z
n
Z
n
=
0
,
1
,
2
,
…
G
n
a
G
a
k
=
e
n
k
a
k
=
e
k
=
n
q
+
r
0
≤
r
<
n
e
=
a
k
=
a
n
q
+
r
=
a
n
q
a
r
=
e
a
r
=
a
r
m
a
m
=
e
n
r
=
0
n
k
k
=
n
s
s
a
k
=
a
n
s
=
(
a
n
)
s
=
e
s
=
e
G
n
a
∈
G
b
=
a
k
b
n
/
d
d
=
gcd
(
k
,
n
)
m
e
=
b
m
=
a
k
m
m
n
k
m
n
/
d
m
(
k
/
d
)
d
n
k
n
/
d
k
/
d
n
/
d
m
(
k
/
d
)
m
m
n
/
d
Z
n
r
1
≤
r
<
n
gcd
(
r
,
n
)
=
1
Z
16
1
3
5
7
9
11
13
15
Z
16
16
Z
16
1
⋅
9
=
9
2
⋅
9
=
2
3
⋅
9
=
11
4
⋅
9
=
4
5
⋅
9
=
13
6
⋅
9
=
6
7
⋅
9
=
15
8
⋅
9
=
8
9
⋅
9
=
1
10
⋅
9
=
10
11
⋅
9
=
3
12
⋅
9
=
12
13
⋅
9
=
5
14
⋅
9
=
14
15
⋅
9
=
7
C
=
{
a
+
b
i
:
a
,
b
∈
R
}
i
2
=
−
1
z
=
a
+
b
i
a
z
b
z
z
=
a
+
b
i
w
=
c
+
d
i
z
+
w
=
(
a
+
b
i
)
+
(
c
+
d
i
)
=
(
a
+
c
)
+
(
b
+
d
)
i
i
2
=
−
1
z
w
(
a
+
b
i
)
(
c
+
d
i
)
=
a
c
+
b
d
i
2
+
a
d
i
+
b
c
i
=
(
a
c
−
b
d
)
+
(
a
d
+
b
c
)
i
z
=
a
+
b
i
z
−
1
∈
C
∗
z
z
−
1
=
z
−
1
z
=
1
z
=
a
+
b
i
z
−
1
=
a
−
b
i
a
2
+
b
2
z
=
a
+
b
i
z
¯
=
a
−
b
i
z
=
a
+
b
i
|
z
|
=
a
2
+
b
2
z
=
2
+
3
i
w
=
1
−
2
i
z
+
w
=
(
2
+
3
i
)
+
(
1
−
2
i
)
=
3
+
i
z
w
=
(
2
+
3
i
)
(
1
−
2
i
)
=
8
−
i
z
−
1
=
2
13
−
3
13
i
|
z
|
=
13
z
¯
=
2
−
3
i
z
=
a
+
b
i
x
y
a
x
b
y
z
1
=
2
+
3
i
z
2
=
1
−
2
i
z
3
=
−
3
+
2
i
θ
x
r
z
=
a
+
b
i
=
r
(
cos
θ
+
i
sin
θ
)
r
=
|
z
|
=
a
2
+
b
2
a
=
r
cos
θ
b
=
r
sin
θ
r
(
cos
θ
+
i
sin
θ
)
r
cis
θ
cos
θ
+
i
sin
θ
z
0
∘
≤
θ
<
360
∘
0
≤
θ
<
2
π
z
=
2
cis
60
∘
a
=
2
cos
60
∘
=
1
b
=
2
sin
60
∘
=
3
z
=
1
+
3
i
z
=
3
2
−
3
2
i
r
=
a
2
+
b
2
=
36
=
6
θ
=
arctan
(
b
a
)
=
arctan
(
−
1
)
=
315
∘
3
2
−
3
2
i
=
6
cis
315
∘
z
=
r
cis
θ
w
=
s
cis
ϕ
z
w
=
r
s
cis
(
θ
+
ϕ
)
z
=
3
cis
(
π
/
3
)
w
=
2
cis
(
π
/
6
)
z
w
=
6
cis
(
π
/
2
)
=
6
i
z
=
r
cis
θ
[
r
cis
θ
]
n
=
r
n
cis
(
n
θ
)
n
=
1
,
2
,
…
n
n
=
1
k
1
≤
k
≤
n
z
n
+
1
=
z
n
z
=
r
n
(
cos
n
θ
+
i
sin
n
θ
)
r
(
cos
θ
+
i
sin
θ
)
=
r
n
+
1
[
(
cos
n
θ
cos
θ
−
sin
n
θ
sin
θ
)
+
i
(
sin
n
θ
cos
θ
+
cos
n
θ
sin
θ
)
]
=
r
n
+
1
[
cos
(
n
θ
+
θ
)
+
i
sin
(
n
θ
+
θ
)
]
=
r
n
+
1
[
cos
(
n
+
1
)
θ
+
i
sin
(
n
+
1
)
θ
]
z
=
1
+
i
z
10
(
1
+
i
)
10
z
10
z
10
=
(
1
+
i
)
10
=
(
2
cis
(
π
4
)
)
10
=
(
2
)
10
cis
(
5
π
2
)
=
32
cis
(
π
2
)
=
32
i
C
∗
Q
∗
R
∗
C
∗
T
=
{
z
∈
C
:
|
z
|
=
1
}
C
∗
H
=
{
1
,
−
1
,
i
,
−
i
}
H
1
−
1
i
−
i
z
4
=
1
z
n
=
1
n
n
z
n
=
1
n
z
=
cis
(
2
k
π
n
)
k
=
0
,
1
,
…
,
n
−
1
n
T
n
z
n
=
cis
(
n
2
k
π
n
)
=
cis
(
2
k
π
)
=
1
z
2
k
π
/
n
2
π
z
n
=
1
n
n
n
T
n
n
n
ω
=
2
2
+
2
2
i
ω
3
=
−
2
2
+
2
2
i
ω
5
=
−
2
2
−
2
2
i
ω
7
=
2
2
−
2
2
i
2
2
2
8
2
2
1,000,000
2
37,398,332
(
mod
46,389
)
0
46,388
n
a
2
a
=
2
k
1
+
2
k
2
+
⋯
+
2
k
n
k
1
<
k
2
<
⋯
<
k
n
a
57
=
2
0
+
2
3
+
2
4
+
2
5
Z
n
b
≡
a
x
(
mod
n
)
c
≡
a
y
(
mod
n
)
b
c
≡
a
x
+
y
(
mod
n
)
a
2
k
(
mod
n
)
k
a
2
0
(
mod
n
)
a
2
1
(
mod
n
)
⋮
a
2
k
(
mod
n
)
n
271
321
(
mod
481
)
321
=
2
0
+
2
6
+
2
8
;
271
321
(
mod
481
)
271
2
0
+
2
6
+
2
8
≡
271
2
0
⋅
271
2
6
⋅
271
2
8
(
mod
481
)
271
2
i
(
mod
481
)
i
=
0
,
6
,
8
271
2
1
=
73,441
≡
329
(
mod
481
)
271
2
2
(
mod
481
)
271
2
2
≡
(
271
2
1
)
2
(
mod
481
)
≡
(
329
)
2
(
mod
481
)
≡
108,241
(
mod
481
)
≡
16
(
mod
481
)
(
a
2
n
)
2
≡
a
2
⋅
2
n
≡
a
2
n
+
1
(
mod
n
)
271
2
6
≡
419
(
mod
481
)
271
2
8
≡
16
(
mod
481
)
271
321
≡
271
2
0
+
2
6
+
2
8
(
mod
481
)
≡
271
2
0
⋅
271
2
6
⋅
271
2
8
(
mod
481
)
≡
271
⋅
419
⋅
16
(
mod
481
)
≡
1,816,784
(
mod
481
)
≡
47
(
mod
481
)
n
3
U
(
20
)
5
U
(
23
)
Z
8
5
th
15
40
(
mod
23
)
Z
60
U
(
8
)
Q
G
G
5
∈
Z
12
3
∈
R
3
∈
R
∗
−
i
∈
C
∗
72
∈
Z
240
312
∈
Z
471
12
10
Z
7
Z
24
15
Z
12
Z
60
Z
13
Z
48
U
(
20
)
U
(
18
)
R
∗
7
C
∗
i
i
2
=
−
1
C
∗
2
i
C
∗
(
1
+
i
)
/
2
C
∗
(
1
+
3
i
)
/
2
7
Z
=
{
…
,
−
7
,
0
,
7
,
14
,
…
}
{
0
,
3
,
6
,
9
,
12
,
15
,
18
,
21
}
{
0
}
{
0
,
6
}
{
0
,
4
,
8
}
{
0
,
3
,
6
,
9
}
{
0
,
2
,
4
,
6
,
8
,
10
}
{
1
,
3
,
7
,
9
}
{
1
,
−
1
,
i
,
−
i
}
G
L
2
(
R
)
(
0
1
−
1
0
)
(
0
1
/
3
3
0
)
(
1
−
1
1
0
)
(
1
−
1
0
1
)
(
1
−
1
−
1
0
)
(
3
/
2
1
/
2
−
1
/
2
3
/
2
)
(
1
0
0
1
)
,
(
−
1
0
0
−
1
)
,
(
0
−
1
1
0
)
,
(
0
1
−
1
0
)
(
1
0
0
1
)
,
(
1
−
1
1
0
)
,
(
−
1
1
−
1
0
)
,
(
0
1
−
1
1
)
,
(
0
−
1
1
−
1
)
,
(
−
1
0
0
−
1
)
Z
18
D
4
Q
8
U
(
30
)
Z
32
∗
Z
Q
∗
R
∗
0
1
,
−
1
a
24
=
e
G
a
1
,
2
,
3
,
4
,
6
,
8
,
12
,
24
n
n
≤
20
U
(
n
)
A
=
(
0
1
−
1
0
)
and
B
=
(
0
−
1
1
−
1
)
G
L
2
(
R
)
A
B
A
B
(
3
−
2
i
)
+
(
5
i
−
6
)
(
4
−
5
i
)
−
(
4
i
−
4
)
¯
(
5
−
4
i
)
(
7
+
2
i
)
(
9
−
i
)
(
9
−
i
)
¯
i
45
(
1
+
i
)
+
(
1
+
i
)
¯
−
3
+
3
i
43
−
18
i
i
a
+
b
i
2
cis
(
π
/
6
)
5
cis
(
9
π
/
4
)
3
cis
(
π
)
cis
(
7
π
/
4
)
/
2
3
+
i
−
3
1
−
i
−
5
2
+
2
i
3
+
i
−
3
i
2
i
+
2
3
2
cis
(
7
π
/
4
)
2
2
cis
(
π
/
4
)
3
cis
(
3
π
/
2
)
(
1
+
i
)
−
1
(
1
−
i
)
6
(
3
+
i
)
5
(
−
i
)
10
(
(
1
−
i
)
/
2
)
4
(
−
2
−
2
i
)
12
(
−
2
+
2
i
)
−
5
(
1
−
i
)
/
2
16
(
i
−
3
)
−
1
/
4
|
z
|
=
|
z
¯
|
z
z
¯
=
|
z
|
2
z
−
1
=
z
¯
/
|
z
|
2
|
z
+
w
|
≤
|
z
|
+
|
w
|
|
z
−
w
|
≥
|
|
z
|
−
|
w
|
|
|
z
w
|
=
|
z
|
|
w
|
292
3171
(
mod
582
)
2557
341
(
mod
5681
)
2071
9521
(
mod
4724
)
971
321
(
mod
765
)
292
1523
a
,
b
∈
G
a
a
−
1
g
∈
G
|
a
|
=
|
g
−
1
a
g
|
a
b
b
a
p
q
Z
p
q
p
r
Z
p
r
Z
p
p
g
h
15
16
G
⟨
g
⟩
∩
⟨
h
⟩
|
⟨
g
⟩
∩
⟨
h
⟩
|
=
1
a
G
⟨
a
m
⟩
∩
⟨
a
n
⟩
Z
n
n
>
2
G
a
b
∈
G
|
a
|
=
m
|
b
|
=
n
gcd
(
m
,
n
)
=
1
⟨
a
⟩
∩
⟨
b
⟩
=
{
e
}
G
G
G
g
,
h
∈
G
m
n
(
g
−
1
)
m
=
e
(
g
h
)
m
n
=
e
G
G
G
n
x
y
=
x
k
gcd
(
k
,
n
)
=
1
y
G
G
2
G
4
G
p
q
gcd
(
p
,
q
)
=
1
G
a
b
p
q
G
Z
n
Z
n
=
0
,
1
,
2
,
…
Z
n
r
1
≤
r
<
n
gcd
(
r
,
n
)
=
1
G
G
g
G
g
G
⟨
g
⟩
G
G
G
m
d
∣
m
G
d
n
−
1
n
z
=
r
(
cos
θ
+
i
sin
θ
)
w
=
s
(
cos
ϕ
+
i
sin
ϕ
)
z
w
=
r
s
[
cos
(
θ
+
ϕ
)
+
i
sin
(
θ
+
ϕ
)
]
C
∗
n
T
n
α
∈
T
α
m
=
1
α
n
=
1
α
d
=
1
d
=
gcd
(
m
,
n
)
z
∈
C
∗
|
z
|
1
z
z
=
cos
θ
+
i
sin
θ
T
θ
∈
Q
z
2
a
x
(
mod
n
)
n
x
Z
3
Z
Z
n
Z
14
1
12
1
12
12
T
n
14
r
C
∗
1
θ
2
π
14
n
U
(
n
)
40
40
U
(
40
)
7
U
U
7
7
U
(
40
)
U
(
49
)
U
(
49
)
U
(
35
)
16
U
(
35
)
U
(
n
)
n
△
A
B
C
S
=
{
A
,
B
,
C
}
π
:
S
→
S
(
A
B
C
A
B
C
)
(
A
B
C
C
A
B
)
(
A
B
C
B
C
A
)
(
A
B
C
A
C
B
)
(
A
B
C
C
B
A
)
(
A
B
C
B
A
C
)
(
A
B
C
B
C
A
)
A
B
B
C
C
A
A
↦
B
B
↦
C
C
↦
A
X
S
X
X
X
=
{
1
,
2
,
…
,
n
}
S
n
S
X
n
S
n
n
n
S
n
n
!
S
n
1
1
2
2
…
n
n
f
:
S
n
→
S
n
f
−
1
f
|
S
n
|
=
n
!
S
n
G
S
5
i
d
σ
=
(
1
2
3
4
5
1
2
3
5
4
)
τ
=
(
1
2
3
4
5
3
2
1
4
5
)
μ
=
(
1
2
3
4
5
3
2
1
5
4
)
G
∘
i
d
σ
τ
μ
i
d
i
d
σ
τ
μ
σ
σ
i
d
μ
τ
τ
τ
μ
i
d
σ
μ
μ
τ
σ
i
d
σ
τ
X
σ
τ
(
σ
∘
τ
)
(
x
)
=
σ
(
τ
(
x
)
)
τ
σ
σ
τ
τ
σ
σ
τ
(
x
)
σ
(
τ
(
x
)
)
σ
(
x
)
(
x
)
σ
σ
=
(
1
2
3
4
4
1
2
3
)
τ
=
(
1
2
3
4
2
1
4
3
)
σ
τ
=
(
1
2
3
4
1
4
3
2
)
τ
σ
=
(
1
2
3
4
3
2
1
4
)
σ
∈
S
X
k
a
1
,
a
2
,
…
,
a
k
∈
X
σ
(
a
1
)
=
a
2
σ
(
a
2
)
=
a
3
⋮
σ
(
a
k
)
=
a
1
σ
(
x
)
=
x
x
∈
X
(
a
1
,
a
2
,
…
,
a
k
)
σ
k
σ
=
(
1
2
3
4
5
6
7
6
3
5
1
4
2
7
)
=
(
1
6
2
3
5
4
)
6
τ
=
(
1
2
3
4
5
6
1
4
2
3
5
6
)
=
(
2
4
3
)
3
(
1
2
3
4
5
6
2
4
1
3
6
5
)
=
(
1
2
4
3
)
(
5
6
)
4
σ
=
(
1
3
5
2
)
and
τ
=
(
2
5
6
)
σ
1
↦
3
,
3
↦
5
,
5
↦
2
,
2
↦
1
τ
2
↦
5
,
5
↦
6
,
6
↦
2
σ
τ
τ
σ
1
↦
3
,
3
↦
5
,
5
↦
6
,
6
↦
2
↦
1
σ
τ
=
(
1
3
5
6
)
μ
=
(
1634
)
σ
μ
=
(
1
6
5
2
)
(
3
4
)
S
X
σ
=
(
a
1
,
a
2
,
…
,
a
k
)
τ
=
(
b
1
,
b
2
,
…
,
b
l
)
a
i
b
j
i
j
(
1
3
5
)
(
2
7
)
(
1
3
5
)
(
3
4
7
)
(
1
3
5
)
(
2
7
)
=
(
1
3
5
)
(
2
7
)
(
1
3
5
)
(
3
4
7
)
=
(
1
3
4
7
5
)
σ
τ
S
X
σ
τ
=
τ
σ
σ
=
(
a
1
,
a
2
,
…
,
a
k
)
τ
=
(
b
1
,
b
2
,
…
,
b
l
)
σ
τ
(
x
)
=
τ
σ
(
x
)
x
∈
X
x
{
a
1
,
a
2
,
…
,
a
k
}
{
b
1
,
b
2
,
…
,
b
l
}
σ
τ
x
σ
(
x
)
=
x
τ
(
x
)
=
x
σ
τ
(
x
)
=
σ
(
τ
(
x
)
)
=
σ
(
x
)
=
x
=
τ
(
x
)
=
τ
(
σ
(
x
)
)
=
τ
σ
(
x
)
x
∈
{
a
1
,
a
2
,
…
,
a
k
}
σ
(
a
i
)
=
a
(
i
mod
k
)
+
1
a
1
↦
a
2
a
2
↦
a
3
⋮
a
k
−
1
↦
a
k
a
k
↦
a
1
τ
(
a
i
)
=
a
i
σ
τ
σ
τ
(
a
i
)
=
σ
(
τ
(
a
i
)
)
=
σ
(
a
i
)
=
a
(
i
mod
k
)
+
1
=
τ
(
a
(
i
mod
k
)
+
1
)
=
τ
(
σ
(
a
i
)
)
=
τ
σ
(
a
i
)
x
∈
{
b
1
,
b
2
,
…
,
b
l
}
σ
τ
S
n
X
=
{
1
,
2
,
…
,
n
}
σ
∈
S
n
X
1
{
σ
(
1
)
,
σ
2
(
1
)
,
…
}
X
1
X
i
X
X
1
X
2
{
σ
(
i
)
,
σ
2
(
i
)
,
…
}
X
2
X
3
,
X
4
,
…
X
r
σ
i
σ
i
(
x
)
=
{
σ
(
x
)
x
∈
X
i
x
x
∉
X
i
σ
=
σ
1
σ
2
⋯
σ
r
X
1
,
X
2
,
…
,
X
r
σ
1
,
σ
2
,
…
,
σ
r
σ
=
(
1
2
3
4
5
6
6
4
3
1
5
2
)
τ
=
(
1
2
3
4
5
6
3
2
1
5
6
4
)
σ
=
(
1624
)
τ
=
(
13
)
(
456
)
σ
τ
=
(
1
3
6
)
(
2
4
5
)
τ
σ
=
(
1
4
3
)
(
2
5
6
)
(
1
)
2
(
a
1
,
a
2
,
…
,
a
n
)
=
(
a
1
a
n
)
(
a
1
a
n
−
1
)
⋯
(
a
1
a
3
)
(
a
1
a
2
)
(
1
6
)
(
2
5
3
)
=
(
1
6
)
(
2
3
)
(
2
5
)
=
(
1
6
)
(
4
5
)
(
2
3
)
(
4
5
)
(
2
5
)
(
1
2
)
(
1
2
)
(
1
3
)
(
2
4
)
(
1
3
)
(
2
4
)
(
1
6
)
(
2
3
)
(
1
6
)
(
2
3
)
(
3
5
)
(
1
6
)
(
1
3
)
(
1
6
)
(
1
3
)
(
3
5
)
(
5
6
)
(
1
6
)
r
i
d
=
τ
1
τ
2
⋯
τ
r
r
r
r
>
1
r
=
2
r
>
2
τ
r
−
1
τ
r
(
a
b
)
(
a
b
)
=
i
d
(
b
c
)
(
a
b
)
=
(
a
c
)
(
b
c
)
(
c
d
)
(
a
b
)
=
(
a
b
)
(
c
d
)
(
a
c
)
(
a
b
)
=
(
a
b
)
(
b
c
)
a
b
c
d
τ
r
−
1
τ
r
i
d
=
τ
1
τ
2
⋯
τ
r
−
3
τ
r
−
2
r
−
2
r
τ
r
−
1
τ
r
r
a
τ
r
−
2
τ
r
−
1
r
−
2
r
a
τ
r
−
2
r
−
2
τ
r
−
3
τ
r
−
2
a
a
r
−
2
σ
σ
σ
σ
σ
=
σ
1
σ
2
⋯
σ
m
=
τ
1
τ
2
⋯
τ
n
m
n
σ
σ
m
⋯
σ
1
i
d
=
σ
σ
m
⋯
σ
1
=
τ
1
⋯
τ
n
σ
m
⋯
σ
1
n
σ
S
n
A
n
n
A
n
n
A
n
S
n
A
n
A
n
σ
σ
=
σ
1
σ
2
⋯
σ
r
σ
i
r
σ
−
1
=
σ
r
σ
r
−
1
⋯
σ
1
A
n
S
n
n
≥
2
A
n
n
!
/
2
A
n
S
n
B
n
σ
S
n
n
≥
2
σ
λ
σ
:
A
n
→
B
n
λ
σ
(
τ
)
=
σ
τ
λ
σ
(
τ
)
=
λ
σ
(
μ
)
σ
τ
=
σ
μ
τ
=
σ
−
1
σ
τ
=
σ
−
1
σ
μ
=
μ
λ
σ
λ
σ
A
4
S
4
A
4
(
1
)
(
12
)
(
34
)
(
13
)
(
24
)
(
14
)
(
23
)
(
123
)
(
132
)
(
124
)
(
142
)
(
134
)
(
143
)
(
234
)
(
243
)
A
4
n
n
n
=
3
,
4
,
…
n
D
n
n
1
,
2
,
…
,
n
n
k
k
+
1
k
−
1
2
n
n
n
D
n
S
n
2
n
D
n
n
≥
3
r
s
r
n
=
1
s
2
=
1
s
r
s
=
r
−
1
n
n
i
d
,
360
∘
n
,
2
⋅
360
∘
n
,
…
,
(
n
−
1
)
⋅
360
∘
n
360
∘
/
n
r
r
r
k
=
k
⋅
360
∘
n
n
n
s
1
,
s
2
,
…
,
s
n
s
k
k
n
s
1
=
s
n
/
2
+
1
,
s
2
=
s
n
/
2
+
2
,
…
,
s
n
/
2
=
s
n
s
1
,
s
2
,
…
,
s
n
s
k
s
=
s
1
s
2
=
1
r
n
=
1
t
n
k
k
+
1
k
−
1
k
+
1
t
=
r
k
k
−
1
t
=
r
k
s
r
s
D
n
D
n
r
s
D
n
=
{
1
,
r
,
r
2
,
…
,
r
n
−
1
,
s
,
r
s
,
r
2
s
,
…
,
r
n
−
1
s
}
s
r
s
=
r
−
1
n
D
4
1
2
3
4
r
=
(
1234
)
r
2
=
(
13
)
(
24
)
r
3
=
(
1432
)
r
4
=
(
1
)
s
1
=
(
24
)
s
2
=
(
13
)
D
4
8
r
s
1
=
(
12
)
(
34
)
r
3
s
1
=
(
14
)
(
23
)
D
4
n
6
6
⋅
4
=
24
24
S
4
24
S
4
1
2
3
4
S
4
180
∘
S
4
S
4
S
4
(
1
3
4
)
(
3
5
4
)
A
3
(
1
2
3
4
5
2
4
1
5
3
)
(
1
2
3
4
5
4
2
5
1
3
)
(
1
2
3
4
5
3
5
1
4
2
)
(
1
2
3
4
5
1
4
3
2
5
)
(
12453
)
(
13
)
(
25
)
(
1345
)
(
234
)
(
12
)
(
1253
)
(
143
)
(
23
)
(
24
)
(
1423
)
(
34
)
(
56
)
(
1324
)
(
1254
)
(
13
)
(
25
)
(
1254
)
(
13
)
(
25
)
2
(
1254
)
−
1
(
123
)
(
45
)
(
1254
)
(
1254
)
2
(
123
)
(
45
)
(
123
)
(
45
)
(
1254
)
−
2
(
1254
)
100
|
(
1254
)
|
|
(
1254
)
2
|
(
12
)
−
1
(
12537
)
−
1
[
(
12
)
(
34
)
(
12
)
(
47
)
]
−
1
[
(
1235
)
(
467
)
]
−
1
(
135
)
(
24
)
(
14
)
(
23
)
(
1324
)
(
134
)
(
25
)
(
17352
)
(
14356
)
(
156
)
(
234
)
(
1426
)
(
142
)
(
17254
)
(
1423
)
(
154632
)
(
142637
)
(
16
)
(
15
)
(
13
)
(
14
)
(
16
)
(
14
)
(
12
)
(
a
1
,
a
2
,
…
,
a
n
)
−
1
(
a
1
,
a
2
,
…
,
a
n
)
−
1
=
(
a
1
,
a
n
,
a
n
−
1
,
…
,
a
2
)
S
4
{
σ
∈
S
4
:
σ
(
1
)
=
3
}
{
σ
∈
S
4
:
σ
(
2
)
=
2
}
{
σ
∈
S
4
:
σ
(
1
)
=
3
σ
(
2
)
=
2
}
S
4
{
(
13
)
,
(
13
)
(
24
)
,
(
132
)
,
(
134
)
,
(
1324
)
,
(
1342
)
}
A
4
S
7
A
7
A
10
15
(
12345
)
(
678
)
A
8
26
S
n
n
=
3
,
…
,
10
A
5
A
6
(
1
)
,
(
a
1
,
a
2
)
(
a
3
,
a
4
)
,
(
a
1
,
a
2
,
a
3
)
,
(
a
1
,
a
2
,
a
3
,
a
4
,
a
5
)
A
5
σ
∈
S
n
n
i
j
σ
i
=
σ
j
i
≡
j
(
mod
n
)
σ
=
σ
1
⋯
σ
m
∈
S
n
σ
σ
1
,
…
,
σ
m
D
5
r
s
r
s
(
12
)
(
34
)
A
4
S
n
n
≥
3
(
123
)
(
12
)
(
12
)
(
123
)
A
n
n
≥
4
D
n
n
≥
3
σ
∈
S
n
σ
n
−
1
σ
∈
S
n
σ
σ
n
−
2
σ
σ
σ
σ
2
3
A
n
n
≥
3
3
(
a
b
)
(
b
c
)
(
a
b
)
(
c
d
)
S
n
(
1
2
)
,
(
13
)
,
…
,
(
1
n
)
(
1
2
)
,
(
23
)
,
…
,
(
n
−
1
,
n
)
(
12
)
,
(
1
2
…
n
)
G
λ
g
:
G
→
G
λ
g
(
a
)
=
g
a
λ
g
G
n
!
n
G
Z
(
G
)
=
{
g
∈
G
:
g
x
=
x
g
for all
x
∈
G
}
D
8
D
10
D
n
τ
=
(
a
1
,
a
2
,
…
,
a
k
)
k
σ
σ
τ
σ
−
1
=
(
σ
(
a
1
)
,
σ
(
a
2
)
,
…
,
σ
(
a
k
)
)
k
μ
k
σ
σ
τ
σ
−
1
=
μ
σ
τ
σ
−
1
(
σ
(
a
i
)
)
=
σ
(
a
i
+
1
)
α
β
S
n
α
∼
β
σ
∈
S
n
σ
α
σ
−
1
=
β
∼
S
n
σ
∈
S
X
σ
n
(
x
)
=
y
n
∈
Z
x
∼
y
∼
X
x
∈
X
σ
∈
S
X
O
x
,
σ
=
{
y
:
x
∼
y
}
{
1
,
2
,
3
,
4
,
5
}
S
5
α
=
(
1254
)
β
=
(
123
)
(
45
)
γ
=
(
13
)
(
25
)
O
x
,
σ
∩
O
y
,
σ
∅
O
x
,
σ
=
O
y
,
σ
σ
∼
H
S
X
x
,
y
∈
X
σ
∈
H
σ
(
x
)
=
y
⟨
σ
⟩
O
x
,
σ
=
X
x
∈
X
α
∈
S
n
n
≥
3
α
β
=
β
α
β
∈
S
n
α
S
n
α
α
−
1
α
σ
∈
A
n
τ
∈
S
n
τ
−
1
σ
τ
∈
A
n
α
−
1
β
−
1
α
β
α
,
β
∈
S
n
r
s
D
n
s
r
s
=
r
−
1
r
k
s
=
s
r
−
k
D
n
r
k
∈
D
n
n
/
gcd
(
k
,
n
)
1
n
S
4
σ
σ
τ
σ
n
n
!
n
2
n
n
n
n
n
!
/
2
S
4
1
4
1
5
8
5
1
6
2
24
S
10
a
3
b
c
a
d
−
1
b
a
,
b
,
c
,
d
G
K
K
L
G
L
L
A
4
A
4
A
4
24
S
8
1
6
S
6
S
10
2
30
N
G
H
G
H
g
∈
G
g
H
=
{
g
h
:
h
∈
H
}
H
g
=
{
h
g
:
h
∈
H
}
H
Z
6
0
3
0
+
H
=
3
+
H
=
{
0
,
3
}
1
+
H
=
4
+
H
=
{
1
,
4
}
2
+
H
=
5
+
H
=
{
2
,
5
}
Z
Z
n
H
S
3
{
(
1
)
,
(
123
)
,
(
132
)
}
H
(
1
)
H
=
(
1
2
3
)
H
=
(
132
)
H
=
{
(
1
)
,
(
1
23
)
,
(
132
)
}
(
1
2
)
H
=
(
1
3
)
H
=
(
2
3
)
H
=
{
(
1
2
)
,
(
1
3
)
,
(
2
3
)
}
H
H
(
1
)
=
H
(
1
2
3
)
=
H
(
132
)
=
{
(
1
)
,
(
1
23
)
,
(
132
)
}
H
(
1
2
)
=
H
(
1
3
)
=
H
(
2
3
)
=
{
(
1
2
)
,
(
1
3
)
,
(
2
3
)
}
K
S
3
{
(
1
)
,
(
1
2
)
}
K
(
1
)
K
=
(
1
2
)
K
=
{
(
1
)
,
(
1
2
)
}
(
1
3
)
K
=
(
1
2
3
)
K
=
{
(
1
3
)
,
(
1
2
3
)
}
(
2
3
)
K
=
(
1
3
2
)
K
=
{
(
2
3
)
,
(
1
3
2
)
}
;
K
K
(
1
)
=
K
(
1
2
)
=
{
(
1
)
,
(
1
2
)
}
K
(
1
3
)
=
K
(
1
3
2
)
=
{
(
1
3
)
,
(
1
3
2
)
}
K
(
2
3
)
=
K
(
1
2
3
)
=
{
(
2
3
)
,
(
1
2
3
)
}
H
G
g
1
,
g
2
∈
G
g
1
H
=
g
2
H
H
g
1
−
1
=
H
g
2
−
1
g
1
H
⊂
g
2
H
g
2
∈
g
1
H
g
1
−
1
g
2
∈
H
H
G
H
G
H
G
G
G
H
G
g
1
H
g
2
H
H
G
g
1
H
∩
g
2
H
=
∅
g
1
H
=
g
2
H
g
1
H
∩
g
2
H
∅
a
∈
g
1
H
∩
g
2
H
a
=
g
1
h
1
=
g
2
h
2
h
1
h
2
H
g
1
=
g
2
h
2
h
1
−
1
g
1
∈
g
2
H
g
1
H
=
g
2
H
G
H
G
H
G
H
G
H
G
[
G
:
H
]
H
G
G
=
Z
6
H
=
{
0
,
3
}
[
G
:
H
]
=
3
G
=
S
3
H
=
{
(
1
)
,
(
123
)
,
(
132
)
}
K
=
{
(
1
)
,
(
12
)
}
[
G
:
H
]
=
2
[
G
:
K
]
=
3
H
G
H
G
H
G
L
H
R
H
H
G
H
G
H
G
ϕ
:
L
H
→
R
H
g
H
∈
L
H
ϕ
(
g
H
)
=
H
g
−
1
ϕ
g
1
H
=
g
2
H
H
g
1
−
1
=
H
g
2
−
1
ϕ
H
g
1
−
1
=
ϕ
(
g
1
H
)
=
ϕ
(
g
2
H
)
=
H
g
2
−
1
g
1
H
=
g
2
H
ϕ
ϕ
(
g
−
1
H
)
=
H
g
H
G
g
∈
G
ϕ
:
H
→
g
H
ϕ
(
h
)
=
g
h
ϕ
H
g
H
ϕ
ϕ
(
h
1
)
=
ϕ
(
h
2
)
h
1
,
h
2
∈
H
h
1
=
h
2
ϕ
(
h
1
)
=
g
h
1
ϕ
(
h
2
)
=
g
h
2
g
h
1
=
g
h
2
h
1
=
h
2
ϕ
g
H
g
h
h
∈
H
ϕ
(
h
)
=
g
h
G
H
G
|
G
|
/
|
H
|
=
[
G
:
H
]
H
G
H
G
G
[
G
:
H
]
|
H
|
|
G
|
=
[
G
:
H
]
|
H
|
G
g
∈
G
g
G
|
G
|
=
p
p
G
g
∈
G
g
e
g
G
g
e
g
|
⟨
g
⟩
|
>
1
p
g
G
p
Z
p
H
K
G
G
⊃
H
⊃
K
[
G
:
K
]
=
[
G
:
H
]
[
H
:
K
]
[
G
:
K
]
=
|
G
|
|
K
|
=
|
G
|
|
H
|
⋅
|
H
|
|
K
|
=
[
G
:
H
]
[
H
:
K
]
A
4
12
6
12
1
2
3
4
6
A
4
6
H
A
4
3
H
3
H
3
6
A
4
6
[
A
4
:
H
]
=
2
H
A
4
H
g
H
=
H
g
g
H
g
−
1
=
H
g
∈
A
4
3
A
4
3
H
(
123
)
H
(
123
)
−
1
=
(
132
)
H
g
h
g
−
1
∈
H
g
∈
A
4
h
∈
H
(
124
)
(
123
)
(
124
)
−
1
=
(
124
)
(
123
)
(
142
)
=
(
243
)
(
243
)
(
123
)
(
243
)
−
1
=
(
243
)
(
123
)
(
234
)
=
(
142
)
H
(
1
)
,
(
123
)
,
(
132
)
,
(
243
)
,
(
243
)
−
1
=
(
234
)
,
(
142
)
,
(
142
)
−
1
=
(
124
)
A
4
6
τ
μ
S
n
σ
∈
S
n
μ
=
σ
τ
σ
−
1
τ
=
(
a
1
,
a
2
,
…
,
a
k
)
μ
=
(
b
1
,
b
2
,
…
,
b
k
)
σ
σ
(
a
1
)
=
b
1
σ
(
a
2
)
=
b
2
⋮
σ
(
a
k
)
=
b
k
μ
=
σ
τ
σ
−
1
τ
=
(
a
1
,
a
2
,
…
,
a
k
)
k
σ
∈
S
n
σ
(
a
i
)
=
b
σ
(
a
(
i
mod
k
)
+
1
)
=
b
′
μ
(
b
)
=
b
′
μ
=
(
σ
(
a
1
)
,
σ
(
a
2
)
,
…
,
σ
(
a
k
)
)
σ
μ
τ
ϕ
ϕ
ϕ
:
N
→
N
ϕ
(
n
)
=
1
n
=
1
n
>
1
ϕ
(
n
)
m
1
≤
m
<
n
gcd
(
m
,
n
)
=
1
U
(
n
)
Z
n
ϕ
(
n
)
|
U
(
12
)
|
=
ϕ
(
12
)
=
4
p
ϕ
(
p
)
=
p
−
1
U
(
n
)
Z
n
|
U
(
n
)
|
=
ϕ
(
n
)
a
n
n
>
0
gcd
(
a
,
n
)
=
1
a
ϕ
(
n
)
≡
1
(
mod
n
)
U
(
n
)
ϕ
(
n
)
a
ϕ
(
n
)
=
1
a
∈
U
(
n
)
a
ϕ
(
n
)
−
1
n
a
ϕ
(
n
)
≡
1
(
mod
n
)
n
=
p
ϕ
(
p
)
=
p
−
1
p
p
∤
a
p
a
a
b
a
p
−
1
≡
1
(
mod
p
)
b
b
p
≡
b
(
mod
p
)
⟨
3
⟩
Z
9
{
(
)
,
(
1
2
)
(
3
4
)
,
(
1
3
)
(
2
4
)
,
(
1
4
)
(
2
3
)
}
S
4
S
4
G
G
p
=
137909
57
137909
(
mod
137909
)
G
g
5
h
7
|
G
|
≥
35
g
h
G
G
60
G
60
⟨
8
⟩
Z
24
⟨
3
⟩
U
(
8
)
3
Z
Z
A
4
S
4
A
n
S
n
D
4
S
4
T
C
∗
H
=
{
(
1
)
,
(
123
)
,
(
132
)
}
S
4
⟨
8
⟩
1
+
⟨
8
⟩
2
+
⟨
8
⟩
3
+
⟨
8
⟩
4
+
⟨
8
⟩
5
+
⟨
8
⟩
6
+
⟨
8
⟩
7
+
⟨
8
⟩
3
Z
1
+
3
Z
2
+
3
Z
S
L
2
(
R
)
G
L
2
(
R
)
S
L
2
(
R
)
G
L
2
(
R
)
n
=
15
a
=
4
4
ϕ
(
15
)
≡
4
8
≡
1
(
mod
15
)
p
=
4
n
+
3
x
2
≡
−
1
(
mod
p
)
H
G
g
1
,
g
2
∈
G
g
1
H
=
g
2
H
H
g
1
−
1
=
H
g
2
−
1
g
1
H
⊂
g
2
H
g
2
∈
g
1
H
g
1
−
1
g
2
∈
H
g
h
g
−
1
∈
H
g
∈
G
h
∈
H
g
H
=
H
g
g
∈
G
g
1
∈
g
H
g
1
∈
H
g
g
H
⊂
H
g
ϕ
:
L
H
→
R
H
ϕ
(
g
H
)
=
H
g
g
n
=
e
g
n
σ
σ
σ
=
(
12
)
(
345
)
(
78
)
(
9
)
(
2
,
3
,
2
,
1
)
(
1
,
2
,
2
,
3
)
α
,
β
∈
S
n
γ
β
=
γ
α
γ
−
1
β
=
γ
α
γ
−
1
γ
∈
S
n
α
β
|
G
|
=
2
n
2
G
[
G
:
H
]
=
2
a
b
H
a
b
∈
H
[
G
:
H
]
=
2
g
H
=
H
g
H
K
G
g
H
∩
g
K
H
∩
K
G
g
(
H
∩
K
)
=
g
H
∩
g
K
H
K
G
∼
G
a
∼
b
h
∈
H
k
∈
K
h
a
k
=
b
H
=
{
(
1
)
,
(
123
)
,
(
132
)
}
A
4
G
n
ϕ
(
n
)
G
n
=
p
1
e
1
p
2
e
2
⋯
p
k
e
k
p
1
,
p
2
,
…
,
p
k
ϕ
(
n
)
=
n
(
1
−
1
p
1
)
(
1
−
1
p
2
)
⋯
(
1
−
1
p
k
)
gcd
(
m
,
n
)
=
1
ϕ
(
m
n
)
=
ϕ
(
m
)
ϕ
(
n
)
n
=
∑
d
∣
n
ϕ
(
d
)
n
A
4
A
4
12
A
4
S
4
S
4
S
4
ϕ
G
m
m
G
G
m
A
4
G
m
391
=
17
⋅
23
a
b
p
100
1000
n
n
−
1
n
100
1000
0
<
n
<
100
1
≤
a
≤
n
7
S
7
7
!
=
5040
10
1
2
5040
n
…
…
A
B
C
B
C
A
B
C
A
B
C
C
C
f
f
−
1
f
f
−
1
f
A
=
00
,
B
=
01
,
…
,
Z
=
25
f
(
p
)
=
p
+
3
mod
26
;
A
↦
D
,
B
↦
E
,
…
,
Z
↦
C
f
−
1
(
p
)
=
p
−
3
mod
26
=
p
+
23
mod
26
3
,
14
,
9
,
7
,
4
,
20
,
3
0
,
11
,
6
,
4
,
1
,
17
,
0
3
26
26
b
f
(
p
)
=
p
+
b
mod
26
E
=
04
S
=
18
18
=
4
+
b
mod
26
b
=
14
f
(
p
)
=
p
+
14
mod
26
f
−
1
(
p
)
=
p
+
12
mod
26
26
f
(
p
)
=
a
p
+
b
mod
26
f
−
1
c
=
a
p
+
b
mod
26
p
a
gcd
(
a
,
26
)
=
1
f
−
1
(
p
)
=
a
−
1
p
−
a
−
1
b
mod
26
f
(
p
)
=
a
p
+
b
mod
26
a
∈
Z
26
gcd
(
a
,
26
)
=
1
a
=
5
gcd
(
5
,
26
)
=
1
a
−
1
=
21
f
(
p
)
=
5
p
+
3
mod
26
3
,
6
,
7
,
23
,
8
,
10
,
3
f
−
1
(
p
)
=
21
p
−
21
⋅
3
mod
26
=
21
p
+
15
mod
26
p
1
p
2
p
=
(
p
1
p
2
)
A
2
×
2
Z
26
f
(
p
)
=
A
p
+
b
b
Z
26
f
−
1
(
p
)
=
A
−
1
p
−
A
−
1
b
7
,
4
,
11
,
15
A
=
(
3
5
1
2
)
A
−
1
=
(
2
21
25
3
)
b
=
(
2
,
2
)
t
f
f
−
1
p
q
n
=
p
q
ϕ
(
n
)
=
m
=
(
p
−
1
)
(
q
−
1
)
ϕ
ϕ
E
m
E
gcd
(
E
,
m
)
=
1
D
D
E
≡
1
(
mod
m
)
n
E
E
n
A
=
00
,
B
=
02
,
…
,
Z
=
25
n
x
y
=
x
E
mod
n
y
x
x
=
y
D
mod
n
D
25
p
=
23
q
=
29
n
=
p
q
=
667
ϕ
(
n
)
=
m
=
(
p
−
1
)
(
q
−
1
)
=
616
E
=
487
gcd
(
616
,
487
)
=
1
25
487
mod
667
=
169
191
E
=
1
+
151
m
(
n
,
D
)
=
(
667
,
191
)
169
191
mod
667
=
25
D
E
≡
1
(
mod
m
)
k
D
E
=
k
m
+
1
=
k
ϕ
(
n
)
+
1
gcd
(
x
,
n
)
=
1
y
D
=
(
x
E
)
D
=
x
D
E
=
x
k
m
+
1
=
(
x
ϕ
(
n
)
)
k
x
=
(
1
)
k
x
=
x
mod
n
x
y
D
mod
n
gcd
(
x
,
n
)
1
n
=
p
q
x
<
n
x
p
q
r
r
<
q
x
=
r
p
gcd
(
x
,
q
)
=
1
m
=
ϕ
(
n
)
=
(
p
−
1
)
(
q
−
1
)
=
ϕ
(
p
)
ϕ
(
q
)
q
x
k
m
=
x
k
ϕ
(
p
)
ϕ
(
q
)
=
(
x
ϕ
(
q
)
)
k
ϕ
(
p
)
=
(
1
)
k
ϕ
(
p
)
=
1
mod
q
t
x
k
m
=
1
+
t
q
y
D
=
x
k
m
+
1
=
x
k
m
x
=
(
1
+
t
q
)
x
=
x
+
t
q
(
r
p
)
=
x
+
t
r
n
=
x
mod
n
D
n
E
n
D
667
=
23
⋅
29
D
(
n
′
,
E
′
)
(
n
′
,
D
′
)
(
n
,
E
)
(
n
,
D
)
x
x
x
′
=
x
D
′
mod
n
′
x
′
x
x
′
x
′
y
′
=
x
′
E
mod
n
ϕ
(
893
456
123
)
7
324
(
mod
895
)
26
!
−
1
2
×
2
A
Z
26
gcd
(
det
(
A
)
,
26
)
=
1
A
=
(
3
4
2
3
)
f
(
p
)
=
A
p
+
b
b
=
(
2
,
5
)
t
x
x
2
x
=
142528
14
25
28
n
=
3551
,
E
=
629
,
x
=
31
n
=
2257
,
E
=
47
,
x
=
23
n
=
120979
,
E
=
13251
,
x
=
142371
n
=
45629
,
E
=
781
,
x
=
231561
2791
112135
25032
442
D
y
n
=
3551
,
D
=
1997
,
y
=
2791
n
=
5893
,
D
=
81
,
y
=
34
n
=
120979
,
D
=
27331
,
y
=
112135
n
=
79403
,
D
=
671
,
y
=
129381
31
14
(
n
,
E
)
D
(
n
,
E
)
=
(
451
,
231
)
(
n
,
E
)
=
(
3053
,
1921
)
(
n
,
E
)
=
(
37986733
,
12371
)
(
n
,
E
)
=
(
16394854313
,
34578451
)
n
=
11
⋅
41
n
=
8779
⋅
4327
n
n
n
E
X
X
E
≡
X
(
mod
n
)
10
15
(
n
,
E
)
D
n
n
n
d
=
2
,
3
,
…
,
n
n
d
n
n
n
=
a
b
n
n
=
x
2
−
y
2
=
(
x
−
y
)
(
x
+
y
)
x
y
n
=
x
2
−
y
2
n
p
gcd
(
a
,
p
)
=
1
a
p
−
1
≡
1
(
mod
p
)
15
2
15
−
1
≡
2
14
≡
4
(
mod
15
)
17
2
17
−
1
≡
2
16
≡
1
(
mod
17
)
n
2
n
−
1
≡
1
(
mod
n
)
342
811
561
771
631
n
b
gcd
(
b
,
n
)
=
1
b
n
−
1
≡
1
(
mod
n
)
n
b
341
2
3
2000
2000
2000
561
=
3
⋅
11
⋅
17
25
×
10
9
n
21
ϕ
p
q
(
p
−
1
)
(
q
−
1
)
m
m
m
m
m
m
n
E
D
D
D
m
128
4
=
268
,
435
,
456
n
n
10
12
E
n
E
D
2
⋅
10
12
E
n
E
D
1
n
m
n
n
n
n
n
(
x
1
,
x
2
,
…
,
x
n
)
3
n
(
x
1
,
x
2
,
…
,
x
n
)
↦
(
x
1
,
x
2
,
…
,
x
n
,
x
1
,
x
2
,
…
,
x
n
,
x
1
,
x
2
,
…
,
x
n
)
i
i
(
0110
)
(
0110
0110
0110
)
(
0110
1110
0110
)
(
0110
)
n
n
2
n
n
m
m
3
n
8
2
8
=
256
8
2
7
=
128
A
=
65
10
=
01000001
2
,
B
=
66
10
=
01000010
2
,
C
=
67
10
=
01000011
2
128
00000000
2
=
0
10
,
⋮
01111111
2
=
127
10
0
1
1
A
=
01000001
2
,
B
=
01000010
2
,
C
=
11000011
2
(
0100
0101
)
1
m
8
16
32
0
1
1
1
1
0
1
(
1001
1000
)
0
1
0
(
000
)
1
(
111
)
(
101
)
1
0
(
111
)
000
001
010
011
100
101
110
111
000
0
1
1
2
1
2
2
3
111
3
2
2
1
2
1
1
0
(
000
)
(
111
)
3
p
q
p
q
n
0
1
p
q
=
1
−
p
1
1
p
0
q
n
p
n
p
=
0.999
(
0.999
)
10
,
000
≈
0.00005
n
(
x
1
,
…
,
x
n
)
p
k
(
n
k
)
q
k
p
n
−
k
k
k
q
n
−
k
p
n
q
k
p
n
−
k
k
(
n
k
)
=
n
!
k
!
(
n
−
k
)
!
n
k
q
k
p
n
−
k
(
n
k
)
q
k
p
n
−
k
p
=
0.995
500
p
n
=
(
0.995
)
500
≈
0.082
(
n
1
)
q
p
n
−
1
=
500
(
0.005
)
(
0.995
)
499
≈
0.204
(
n
2
)
q
2
p
n
−
2
=
500
⋅
499
2
(
0.005
)
2
(
0.995
)
498
≈
0.257
1
−
0.082
−
0.204
−
0.257
=
0.457
(
n
,
m
)
m
n
(
n
,
m
)
E
:
Z
2
m
→
Z
2
n
D
:
Z
2
n
→
Z
2
m
E
E
D
(
8
,
7
)
E
(
x
7
,
x
6
,
…
,
x
1
)
=
(
x
8
,
x
7
,
…
,
x
1
)
x
8
=
x
7
+
x
6
+
⋯
+
x
1
Z
2
x
=
(
x
1
,
…
,
x
n
)
y
=
(
y
1
,
…
,
y
n
)
n
d
(
x
,
y
)
x
y
x
y
d
min
d
(
x
,
y
)
x
y
w
(
x
)
x
1
x
w
(
x
)
=
d
(
x
,
0
)
0
=
(
00
⋯
0
)
x
y
x
x
=
(
10101
)
y
=
(
11010
)
z
=
(
00011
)
C
d
(
x
,
y
)
=
4
,
d
(
x
,
z
)
=
3
,
d
(
y
,
z
)
=
3
w
(
x
)
=
3
,
w
(
y
)
=
3
,
w
(
z
)
=
2
x
y
z
n
w
(
x
)
=
d
(
x
,
0
)
d
(
x
,
y
)
≥
0
d
(
x
,
y
)
=
0
x
=
y
d
(
x
,
y
)
=
d
(
y
,
x
)
d
(
x
,
y
)
≤
d
(
x
,
z
)
+
d
(
z
,
y
)
x
=
(
1101
)
y
=
(
1100
)
(
1101
)
(
1100
)
(
1100
)
d
(
x
,
y
)
=
1
x
=
(
1100
)
y
=
(
1010
)
d
(
x
,
y
)
=
2
x
y
2
0000
0011
0101
0110
1001
1010
1100
1111
0000
0
2
2
2
2
2
2
4
0011
2
0
2
2
2
2
4
2
0101
2
2
0
2
2
4
2
2
0110
2
2
2
0
4
2
2
2
1001
2
2
2
4
0
2
2
2
1010
2
2
4
2
2
0
2
2
1100
2
4
2
2
2
2
0
2
1111
4
2
2
2
2
2
2
0
x
y
d
(
x
,
y
)
=
1
x
y
x
y
y
d
(
x
,
y
)
=
2
x
y
d
min
=
2
d
(
x
,
y
)
=
2
y
z
d
(
x
,
z
)
=
d
(
y
,
z
)
=
1
x
y
d
min
≥
3
x
y
d
(
x
,
y
)
=
1
d
(
z
,
y
)
≥
2
z
x
3
C
d
min
=
2
n
+
1
C
n
2
n
C
x
y
n
d
(
x
,
y
)
≤
n
z
x
2
n
+
1
≤
d
(
x
,
z
)
≤
d
(
x
,
y
)
+
d
(
y
,
z
)
≤
n
+
d
(
y
,
z
)
d
(
y
,
z
)
≥
n
+
1
y
x
x
y
2
n
1
≤
d
(
x
,
y
)
≤
2
n
2
n
+
1
y
1
2
n
c
1
=
(
00000
)
c
2
=
(
00111
)
c
3
=
(
11100
)
c
4
=
(
11011
)
00000
00111
11100
11011
00000
0
3
3
4
00111
3
0
4
3
11100
3
4
0
3
11011
4
3
3
0
(
32
,
6
)
Z
2
n
n
n
0
(
11000101
)
+
(
11000101
)
=
(
00000000
)
(
0000000
)
(
0001111
)
(
0010101
)
(
0011010
)
(
0100110
)
(
0101001
)
(
0110011
)
(
0111100
)
(
1000011
)
(
1001100
)
(
1010110
)
(
1011001
)
(
1100101
)
(
1101010
)
(
1110000
)
(
1111111
)
Z
2
7
d
min
=
3
3
x
y
n
w
(
x
+
y
)
=
d
(
x
,
y
)
x
y
n
x
y
x
y
x
y
1
1
+
1
=
0
0
+
0
=
0
1
+
0
=
1
0
+
1
=
1
d
min
C
d
min
C
d
min
=
min
{
w
(
x
)
:
x
0
}
d
min
=
min
{
d
(
x
,
y
)
:
x
y
}
=
min
{
d
(
x
,
y
)
:
x
+
y
0
}
=
min
{
w
(
x
+
y
)
:
x
+
y
0
}
=
min
{
w
(
z
)
:
z
0
}
3
n
x
⋅
y
=
x
1
y
1
+
⋯
+
x
n
y
n
x
=
(
x
1
,
x
2
,
…
,
x
n
)
t
y
=
(
y
1
,
y
2
,
…
,
y
n
)
t
n
x
=
(
011001
)
t
y
=
(
110101
)
t
x
⋅
y
=
0
x
⋅
y
=
x
t
y
=
(
x
1
x
2
⋯
x
n
)
(
y
1
y
2
⋮
y
n
)
=
x
1
y
1
+
x
2
y
2
+
⋯
+
x
n
y
n
3
3
1
n
x
=
(
x
1
,
x
2
,
…
,
x
n
)
t
1
x
1
+
x
2
+
⋯
+
x
n
=
0
4
x
=
(
x
1
,
x
2
,
x
3
,
x
4
)
t
1
x
1
+
x
2
+
x
3
+
x
4
=
0
x
⋅
1
=
x
t
1
=
(
x
1
x
2
x
3
x
4
)
(
1
1
1
1
)
=
0
M
m
×
n
(
Z
2
)
m
×
n
Z
2
Z
2
H
∈
M
m
×
n
(
Z
2
)
n
x
H
x
=
0
H
Null
(
H
)
m
×
n
Z
2
H
H
=
(
0
1
0
1
0
1
1
1
1
0
0
0
1
1
1
)
5
x
=
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
)
t
H
H
x
=
0
x
2
+
x
4
=
0
x
1
+
x
2
+
x
3
+
x
4
=
0
x
3
+
x
4
+
x
5
=
0
5
(
00000
)
(
11110
)
(
10101
)
(
01011
)
H
M
m
×
n
(
Z
2
)
H
Z
2
n
x
,
y
∈
Null
(
H
)
H
M
m
×
n
(
Z
2
)
H
x
=
0
H
y
=
0
H
(
x
+
y
)
=
H
x
+
H
y
=
0
+
0
=
0
x
+
y
H
H
∈
M
m
×
n
(
Z
2
)
C
H
=
(
0
0
0
1
1
1
0
1
1
0
1
1
1
0
1
0
0
1
)
6
x
=
(
010011
)
t
x
H
x
=
(
0
1
1
)
H
H
H
m
×
n
Z
2
n
>
m
m
m
×
m
I
m
H
=
(
A
∣
I
m
)
A
m
×
(
n
−
m
)
(
a
11
a
12
⋯
a
1
,
n
−
m
a
21
a
22
⋯
a
2
,
n
−
m
⋮
⋮
⋱
⋮
a
m
1
a
m
2
⋯
a
m
,
n
−
m
)
I
m
m
×
m
(
1
0
⋯
0
0
1
⋯
0
⋮
⋮
⋱
⋮
0
0
⋯
1
)
n
×
(
n
−
m
)
G
=
(
I
n
−
m
A
)
x
G
x
=
y
H
y
=
0
x
G
y
(
000
)
,
(
001
)
,
(
010
)
,
…
,
(
111
)
A
=
(
0
1
1
1
1
0
1
0
1
)
G
=
(
1
0
0
0
1
0
0
0
1
0
1
1
1
1
0
1
0
1
)
H
=
(
0
1
1
1
0
0
1
1
0
0
1
0
1
0
1
0
0
1
)
H
6
1
1
x
=
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
,
x
6
)
0
=
H
x
=
(
x
2
+
x
3
+
x
4
x
1
+
x
2
+
x
5
x
1
+
x
3
+
x
6
)
x
2
+
x
3
+
x
4
=
0
x
1
+
x
2
+
x
5
=
0
x
1
+
x
3
+
x
6
=
0
x
4
x
2
x
3
x
5
x
1
x
2
x
6
x
1
x
3
x
4
x
5
x
6
x
1
x
2
x
3
x
4
x
5
x
6
H
(
000000
)
(
001101
)
(
010110
)
(
011011
)
(
100011
)
(
101110
)
(
110101
)
(
111000
)
.
G
x
G
x
000
000000
001
001101
010
010110
011
011011
100
100011
101
101110
110
110101
111
111000
H
∈
M
m
×
n
(
Z
2
)
Null
(
H
)
x
∈
Z
2
n
n
−
m
m
H
x
=
0
m
n
−
m
H
(
n
,
n
−
m
)
n
−
m
x
m
G
n
×
k
C
=
{
y
:
G
x
=
y
for
x
∈
Z
2
k
}
(
n
,
k
)
C
G
x
1
=
y
1
G
x
2
=
y
2
y
1
+
y
2
C
G
(
x
1
+
x
2
)
=
G
x
1
+
G
x
2
=
y
1
+
y
2
G
x
=
G
y
x
=
y
G
x
=
G
y
G
x
−
G
y
=
G
(
x
−
y
)
=
0
k
G
(
x
−
y
)
x
1
−
y
1
,
…
,
x
k
−
y
k
I
k
G
G
(
x
−
y
)
=
0
x
=
y
H
=
(
A
∣
I
m
)
m
×
n
G
=
(
I
n
−
m
A
)
n
×
(
n
−
m
)
H
G
=
0
C
=
H
G
i
j
C
c
i
j
=
∑
k
=
1
n
h
i
k
g
k
j
=
∑
k
=
1
n
−
m
h
i
k
g
k
j
+
∑
k
=
n
−
m
+
1
n
h
i
k
g
k
j
=
∑
k
=
1
n
−
m
a
i
k
δ
k
j
+
∑
k
=
n
−
m
+
1
n
δ
i
−
(
m
−
n
)
,
k
a
k
j
=
a
i
j
+
a
i
j
=
0
δ
i
j
=
{
1
i
=
j
0
i
j
H
=
(
A
∣
I
m
)
m
×
n
G
=
(
I
n
−
m
A
)
n
×
(
n
−
m
)
H
C
G
y
C
H
y
=
0
C
H
y
∈
C
G
x
=
y
x
∈
Z
2
m
H
y
=
H
G
x
=
0
y
=
(
y
1
,
…
,
y
n
)
t
H
x
Z
2
n
−
m
G
x
t
=
y
H
y
=
0
a
11
y
1
+
a
12
y
2
+
⋯
+
a
1
,
n
−
m
y
n
−
m
+
y
n
−
m
+
1
=
0
a
21
y
1
+
a
22
y
2
+
⋯
+
a
2
,
n
−
m
y
n
−
m
+
y
n
−
m
+
1
=
0
⋮
a
m
1
y
1
+
a
m
2
y
2
+
⋯
+
a
m
,
n
−
m
y
n
−
m
+
y
n
−
m
+
1
=
0
y
n
−
m
+
1
,
…
,
y
n
y
1
,
…
,
y
n
−
m
y
n
−
m
+
1
=
a
11
y
1
+
a
12
y
2
+
⋯
+
a
1
,
n
−
m
y
n
−
m
y
n
−
m
+
1
=
a
21
y
1
+
a
22
y
2
+
⋯
+
a
2
,
n
−
m
y
n
−
m
⋮
y
n
−
m
+
1
=
a
m
1
y
1
+
a
m
2
y
2
+
⋯
+
a
m
,
n
−
m
y
n
−
m
x
i
=
y
i
i
=
1
,
…
,
n
−
m
H
e
1
=
(
100
⋯
00
)
t
e
2
=
(
010
⋯
00
)
t
⋮
e
n
=
(
000
⋯
01
)
t
n
Z
2
n
1
m
×
n
H
H
e
i
i
H
(
1
1
1
0
0
1
0
0
1
0
1
1
0
0
1
)
(
0
1
0
0
0
)
=
(
1
0
1
)
e
i
n
1
i
0
H
∈
M
m
×
n
(
Z
2
)
H
e
i
i
H
H
m
×
n
H
H
Null
(
H
)
2
2
e
i
i
=
1
,
…
,
n
H
e
i
i
H
e
i
H
i
H
H
e
i
0
H
1
=
(
1
1
1
0
0
1
0
0
1
0
1
1
0
0
1
)
H
2
=
(
1
1
1
0
0
1
0
0
0
0
1
1
0
0
1
)
H
1
H
2
H
H
2
3
H
=
(
1
1
1
0
1
0
0
1
1
1
0
0
)
H
Null
(
H
)
4
2
(
1100
)
(
1010
)
(
1001
)
(
0110
)
(
0101
)
(
0011
)
Null
(
H
)
H
H
H
H
H
H
H
n
e
i
+
e
j
1
i
j
w
(
e
i
+
e
j
)
=
2
i
j
0
=
H
(
e
i
+
e
j
)
=
H
e
i
+
H
e
j
i
j
H
H
2
3
=
8
(
0
0
0
)
,
(
1
0
0
)
,
(
0
1
0
)
,
(
0
0
1
)
3
H
m
×
n
n
−
m
m
2
m
0
,
e
1
,
…
,
e
m
2
m
−
(
1
+
m
)
n
n
H
=
(
1
1
1
0
0
0
1
0
1
0
1
0
0
0
1
)
5
x
=
(
11011
)
t
y
=
(
01011
)
t
H
x
=
(
0
0
0
)
and
H
y
=
(
1
0
1
)
x
y
x
y
H
y
H
y
0
1
x
H
m
×
n
x
∈
Z
2
n
x
H
x
m
×
n
H
x
n
x
x
=
c
+
e
c
e
H
x
x
e
H
x
=
H
(
c
+
e
)
=
H
c
+
H
e
=
0
+
H
e
=
H
e
H
e
i
H
H
∈
M
m
×
n
(
Z
2
)
H
r
n
r
0
r
H
i
i
H
=
(
1
0
1
1
0
0
0
1
1
0
1
0
1
1
1
0
0
1
)
6
x
=
(
111110
)
t
y
=
(
111111
)
t
z
=
(
010111
)
t
H
x
=
(
1
1
1
)
,
H
y
=
(
1
1
0
)
,
H
z
=
(
1
0
0
)
x
z
x
z
(
110110
)
(
010011
)
y
H
y
C
Z
2
n
C
Z
2
n
C
(
n
,
m
)
C
Z
2
n
x
+
C
x
∈
Z
2
n
2
n
−
m
C
Z
2
n
C
(
5
,
3
)
H
=
(
0
1
1
0
0
1
0
0
1
0
1
1
0
0
1
)
(
00000
)
(
01101
)
(
10011
)
(
11110
)
2
5
−
2
=
2
3
C
Z
2
5
2
2
=
4
C
C
(
00000
)
(
01101
)
(
10011
)
(
11110
)
(
10000
)
+
C
(
10000
)
(
11101
)
(
00011
)
(
01110
)
(
01000
)
+
C
(
01000
)
(
00101
)
(
11011
)
(
10110
)
(
00100
)
+
C
(
00100
)
(
01001
)
(
10111
)
(
11010
)
(
00010
)
+
C
(
00010
)
(
01111
)
(
10001
)
(
11100
)
(
00001
)
+
C
(
00001
)
(
01100
)
(
10010
)
(
11111
)
(
10100
)
+
C
(
00111
)
(
01010
)
(
10100
)
(
11001
)
(
00110
)
+
C
(
00110
)
(
01011
)
(
10101
)
(
11000
)
x
r
n
e
r
=
e
+
x
x
=
e
+
r
r
e
+
C
e
e
n
r
+
e
x
r
=
(
01111
)
r
(
00010
)
+
C
(
01101
)
=
(
01111
)
+
(
00010
)
(
000
)
(
00000
)
(
001
)
(
00001
)
(
010
)
(
00010
)
(
011
)
(
10000
)
(
100
)
(
00100
)
(
101
)
(
01000
)
(
110
)
(
00110
)
(
111
)
(
10100
)
C
(
n
,
k
)
H
x
y
Z
2
n
x
y
C
H
x
=
H
y
n
n
x
y
C
x
−
y
∈
C
H
(
x
−
y
)
=
0
H
x
=
H
y
C
x
=
(
01111
)
H
x
=
(
0
1
0
)
(
00010
)
(
n
,
k
)
2
n
−
k
C
(
32
,
24
)
2
24
2
32
−
24
=
2
8
=
256
d
=
6
56
10
56
10
C
H
=
[
0
1
0
1
0
1
1
1
1
0
0
0
1
1
1
]
x
=
11100
x
C
H
x
x
0
1
2
3
4
5
6
7
8
000
001
010
011
101
110
111
000
001
4
Z
2
4
(
0110
)
(
1001
)
(
1010
)
(
1100
)
(
0000
)
∉
C
n
(
011010
)
,
(
011100
)
(
11110101
)
,
(
01010100
)
(
00110
)
,
(
01111
)
(
1001
)
,
(
0111
)
2
2
n
(
011010
)
(
11110101
)
(
01111
)
(
1011
)
3
4
C
7
C
(
011010
)
(
011100
)
(
110111
)
(
110000
)
(
011100
)
(
011011
)
(
111011
)
(
100011
)
(
000000
)
(
010101
)
(
110100
)
(
110011
)
(
000000
)
(
011100
)
(
110101
)
(
110001
)
(
0110110
)
(
0111100
)
(
1110000
)
(
1111111
)
(
1001001
)
(
1000011
)
(
0001111
)
(
0000000
)
d
min
=
2
d
min
=
1
(
n
,
k
)
(
0
1
0
0
0
1
0
1
0
1
1
0
0
1
0
)
(
1
0
1
0
0
0
1
1
0
1
0
0
0
1
0
0
1
0
1
1
0
0
0
1
)
(
1
0
0
1
1
0
1
0
1
1
)
(
0
0
0
1
1
1
1
0
1
1
0
0
1
1
1
0
1
0
1
0
1
0
1
1
0
0
1
1
)
(
00000
)
,
(
00101
)
,
(
10011
)
,
(
10110
)
G
=
(
0
1
0
0
1
0
0
1
1
1
)
(
000000
)
,
(
010111
)
,
(
101101
)
,
(
111010
)
G
=
(
1
0
0
1
1
0
1
1
0
1
1
1
)
(
5
,
2
)
C
H
=
(
0
1
0
0
1
1
0
1
0
1
0
0
1
1
1
)
01111
10101
01110
00011
1000
p
p
=
0.01
p
=
0.0001
(
1
1
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
0
0
1
)
(
0
1
1
0
0
0
1
1
0
1
0
0
0
1
0
0
1
0
1
1
0
0
0
1
)
(
1
1
1
0
1
0
0
1
)
(
0
0
0
1
0
0
0
0
1
1
0
1
0
0
1
0
1
0
0
1
0
0
1
1
0
0
0
1
)
G
=
(
1
1
0
0
1
)
G
=
(
1
0
0
1
1
1
1
0
)
H
=
(
0
1
1
1
1
0
0
0
1
1
1
0
1
0
1
)
C
Z
2
3
(
000
)
(
111
)
C
Z
2
3
C
(
0
1
0
0
0
1
0
1
0
1
1
0
0
1
0
)
(
0
0
1
0
0
1
1
0
1
0
0
1
0
1
0
1
1
0
0
1
)
(
1
0
0
1
1
0
1
0
1
1
)
(
1
0
0
1
1
1
1
1
1
1
0
0
1
1
1
0
1
0
1
0
1
1
1
1
0
0
1
0
)
C
(
10000
)
+
C
(
01000
)
+
C
(
00100
)
+
C
(
00010
)
+
C
(
11000
)
+
C
(
01100
)
+
C
(
01010
)
+
C
C
x
y
z
n
w
(
x
)
=
d
(
x
,
0
)
d
(
x
,
y
)
=
d
(
x
+
z
,
y
+
z
)
d
(
x
,
y
)
=
w
(
x
−
y
)
X
d
:
X
×
X
→
R
d
(
x
,
y
)
≥
0
x
,
y
∈
X
d
(
x
,
y
)
=
0
x
=
y
d
(
x
,
y
)
=
d
(
y
,
x
)
d
(
x
,
y
)
≤
d
(
x
,
z
)
+
d
(
z
,
y
)
Z
2
n
C
i
C
C
x
∈
C
y
↦
x
+
y
C
20
e
i
n
1
i
0
H
∈
M
m
×
n
(
Z
2
)
H
e
i
i
H
C
(
n
,
k
)
C
C
⊥
=
{
x
∈
Z
2
n
:
x
⋅
y
=
0
for all
y
∈
C
}
C
C
(
1
1
1
0
0
0
0
1
0
1
1
0
0
1
0
)
C
⊥
(
n
,
n
−
k
)
C
C
⊥
H
m
×
n
Z
2
i
i
m
H
=
(
0
0
0
1
1
1
0
1
1
0
0
1
1
0
1
0
1
0
)
i
i
(
101011
)
H
(
101101
)
(
001001
)
(
0010000101
)
(
0000101100
)
(
m
,
n
)
k
0
(
16
,
12
)
(
7
,
4
)
7
7
4
2
4
=
16
7
d
=
3
7
3
(
n
,
k
)
r
r
d
(
d
−
1
)
/
2
r
n
1
+
(
n
1
)
+
(
n
2
)
+
⋯
+
(
n
r
)
d
k
2
k
(
(
n
0
)
+
(
n
1
)
+
(
n
2
)
+
⋯
+
(
n
d
−
1
2
)
)
=
2
n
d
32
Z
4
T
i
(
G
,
⋅
)
(
H
,
∘
)
ϕ
:
G
→
H
ϕ
(
a
⋅
b
)
=
ϕ
(
a
)
∘
ϕ
(
b
)
a
b
G
G
H
G
≅
H
G
H
ϕ
Z
4
≅
⟨
i
⟩
ϕ
:
Z
4
→
⟨
i
⟩
ϕ
(
n
)
=
i
n
ϕ
ϕ
ϕ
(
0
)
=
1
ϕ
(
1
)
=
i
ϕ
(
2
)
=
−
1
ϕ
(
3
)
=
−
i
ϕ
(
m
+
n
)
=
i
m
+
n
=
i
m
i
n
=
ϕ
(
m
)
ϕ
(
n
)
ϕ
(
R
,
+
)
(
R
+
,
⋅
)
ϕ
(
x
+
y
)
=
e
x
+
y
=
e
x
e
y
=
ϕ
(
x
)
ϕ
(
y
)
ϕ
Q
∗
2
n
ϕ
:
Z
→
Q
∗
ϕ
(
n
)
=
2
n
ϕ
(
m
+
n
)
=
2
m
+
n
=
2
m
2
n
=
ϕ
(
m
)
ϕ
(
n
)
ϕ
{
2
n
:
n
∈
Z
}
Q
∗
m
n
ϕ
(
m
)
ϕ
(
n
)
m
>
n
ϕ
(
m
)
=
ϕ
(
n
)
2
m
=
2
n
2
m
−
n
=
1
m
−
n
>
0
Z
8
Z
12
U
(
8
)
≅
U
(
12
)
U
(
8
)
=
{
1
,
3
,
5
,
7
}
U
(
12
)
=
{
1
,
5
,
7
,
11
}
ϕ
:
U
(
8
)
→
U
(
12
)
1
↦
1
3
↦
5
5
↦
7
7
↦
11
ϕ
ψ
ψ
(
1
)
=
1
ψ
(
3
)
=
11
ψ
(
5
)
=
5
ψ
(
7
)
=
7
Z
2
×
Z
2
S
3
Z
6
Z
6
S
3
ϕ
:
Z
6
→
S
3
a
,
b
∈
S
3
a
b
b
a
ϕ
m
n
Z
6
ϕ
(
m
)
=
a
and
ϕ
(
n
)
=
b
a
b
=
ϕ
(
m
)
ϕ
(
n
)
=
ϕ
(
m
+
n
)
=
ϕ
(
n
+
m
)
=
ϕ
(
n
)
ϕ
(
m
)
=
b
a
a
b
ϕ
:
G
→
H
ϕ
−
1
:
H
→
G
|
G
|
=
|
H
|
G
H
G
H
G
n
H
n
ϕ
h
1
h
2
H
ϕ
g
1
,
g
2
∈
G
ϕ
(
g
1
)
=
h
1
ϕ
(
g
2
)
=
h
2
h
1
h
2
=
ϕ
(
g
1
)
ϕ
(
g
2
)
=
ϕ
(
g
1
g
2
)
=
ϕ
(
g
2
g
1
)
=
ϕ
(
g
2
)
ϕ
(
g
1
)
=
h
2
h
1
Z
G
a
G
ϕ
:
Z
→
G
ϕ
:
n
↦
a
n
ϕ
(
m
+
n
)
=
a
m
+
n
=
a
m
a
n
=
ϕ
(
m
)
ϕ
(
n
)
ϕ
m
n
Z
m
n
m
>
n
a
m
a
n
a
m
=
a
n
a
m
−
n
=
e
m
−
n
>
0
a
G
a
n
n
ϕ
(
n
)
=
a
n
G
n
G
Z
n
G
n
a
ϕ
:
Z
n
→
G
ϕ
:
k
↦
a
k
0
≤
k
<
n
ϕ
G
p
p
G
Z
p
G
G
Z
3
Z
3
+
0
1
2
0
0
1
2
1
1
2
0
2
2
0
1
Z
3
G
=
{
(
0
)
,
(
0
1
2
)
,
(
0
2
1
)
}
0
↦
(
0
1
2
0
1
2
)
=
(
0
)
1
↦
(
0
1
2
1
2
0
)
=
(
0
1
2
)
2
↦
(
0
1
2
2
0
1
)
=
(
0
2
1
)
G
G
¯
G
g
∈
G
λ
g
:
G
→
G
λ
g
(
a
)
=
g
a
λ
g
G
λ
g
λ
g
(
a
)
=
λ
g
(
b
)
g
a
=
λ
g
(
a
)
=
λ
g
(
b
)
=
g
b
a
=
b
λ
g
a
∈
G
b
λ
g
(
b
)
=
a
b
=
g
−
1
a
G
¯
G
¯
=
{
λ
g
:
g
∈
G
}
G
¯
G
G
¯
(
λ
g
∘
λ
h
)
(
a
)
=
λ
g
(
h
a
)
=
g
h
a
=
λ
g
h
(
a
)
λ
e
(
a
)
=
e
a
=
a
(
λ
g
−
1
∘
λ
g
)
(
a
)
=
λ
g
−
1
(
g
a
)
=
g
−
1
g
a
=
a
=
λ
e
(
a
)
G
G
¯
ϕ
:
g
↦
λ
g
ϕ
(
g
h
)
=
λ
g
h
=
λ
g
λ
h
=
ϕ
(
g
)
ϕ
(
h
)
ϕ
(
g
)
(
a
)
=
ϕ
(
h
)
(
a
)
g
a
=
λ
g
a
=
λ
h
a
=
h
a
g
=
h
ϕ
ϕ
(
g
)
=
λ
g
λ
g
∈
G
¯
g
↦
λ
g
G
G
H
G
H
G
×
H
G
G
(
G
,
⋅
)
(
H
,
∘
)
G
H
(
g
,
h
)
∈
G
×
H
g
∈
G
h
∈
H
G
×
H
(
g
1
,
h
1
)
(
g
2
,
h
2
)
=
(
g
1
⋅
g
2
,
h
1
∘
h
2
)
;
G
H
⋅
∘
(
g
1
,
h
1
)
(
g
2
,
h
2
)
=
(
g
1
g
2
,
h
1
h
2
)
G
H
G
×
H
(
g
1
,
h
1
)
(
g
2
,
h
2
)
=
(
g
1
g
2
,
h
1
h
2
)
g
1
,
g
2
∈
G
h
1
,
h
2
∈
H
e
G
e
H
G
H
(
e
G
,
e
H
)
G
×
H
(
g
,
h
)
∈
G
×
H
(
g
−
1
,
h
−
1
)
G
H
R
R
R
×
R
=
R
2
(
a
,
b
)
+
(
c
,
d
)
=
(
a
+
c
,
b
+
d
)
(
0
,
0
)
(
a
,
b
)
(
−
a
,
−
b
)
Z
2
×
Z
2
=
{
(
0
,
0
)
,
(
0
,
1
)
,
(
1
,
0
)
,
(
1
,
1
)
}
Z
2
×
Z
2
Z
4
(
a
,
b
)
Z
2
×
Z
2
2
(
a
,
b
)
+
(
a
,
b
)
=
(
0
,
0
)
Z
4
G
×
H
G
H
∏
i
=
1
n
G
i
=
G
1
×
G
2
×
⋯
×
G
n
G
1
,
G
2
,
…
,
G
n
G
=
G
1
=
G
2
=
⋯
=
G
n
G
n
G
1
×
G
2
×
⋯
×
G
n
Z
2
n
n
n
(
01011101
)
+
(
01001011
)
=
(
00010110
)
(
g
,
h
)
∈
G
×
H
g
h
r
s
(
g
,
h
)
G
×
H
r
s
m
r
s
n
=
|
(
g
,
h
)
|
(
g
,
h
)
m
=
(
g
m
,
h
m
)
=
(
e
G
,
e
H
)
(
g
n
,
h
n
)
=
(
g
,
h
)
n
=
(
e
G
,
e
H
)
n
m
n
≤
m
r
s
n
n
r
s
m
r
s
m
≤
n
m
n
(
g
1
,
…
,
g
n
)
∈
∏
G
i
g
i
r
i
G
i
(
g
1
,
…
,
g
n
)
∏
G
i
r
1
,
…
,
r
n
(
8
,
56
)
∈
Z
12
×
Z
60
gcd
(
8
,
12
)
=
4
8
12
/
4
=
3
Z
12
56
Z
60
15
3
15
15
(
8
,
56
)
15
Z
12
×
Z
60
Z
2
×
Z
3
(
0
,
0
)
,
(
0
,
1
)
,
(
0
,
2
)
,
(
1
,
0
)
,
(
1
,
1
)
,
(
1
,
2
)
Z
2
×
Z
2
Z
4
Z
2
×
Z
3
≅
Z
6
Z
2
×
Z
3
(
1
,
1
)
Z
2
×
Z
3
Z
m
×
Z
n
Z
m
n
gcd
(
m
,
n
)
=
1
Z
m
×
Z
n
≅
Z
m
n
gcd
(
m
,
n
)
=
1
gcd
(
m
,
n
)
=
d
>
1
Z
m
×
Z
n
m
n
/
d
m
n
(
a
,
b
)
∈
Z
m
×
Z
n
(
a
,
b
)
+
(
a
,
b
)
+
⋯
+
(
a
,
b
)
⏟
m
n
/
d
times
=
(
0
,
0
)
(
a
,
b
)
Z
m
×
Z
n
lcm
(
m
,
n
)
=
m
n
gcd
(
m
,
n
)
=
1
n
1
,
…
,
n
k
∏
i
=
1
k
Z
n
i
≅
Z
n
1
⋯
n
k
gcd
(
n
i
,
n
j
)
=
1
i
j
m
=
p
1
e
1
⋯
p
k
e
k
p
i
Z
m
≅
Z
p
1
e
1
×
⋯
×
Z
p
k
e
k
p
i
e
i
p
j
e
j
i
j
Z
p
1
e
1
×
⋯
×
Z
p
k
e
k
p
1
,
…
,
p
k
G
H
K
G
=
H
K
=
{
h
k
:
h
∈
H
,
k
∈
K
}
H
∩
K
=
{
e
}
h
k
=
k
h
k
∈
K
h
∈
H
G
H
K
U
(
8
)
H
=
{
1
,
3
}
and
K
=
{
1
,
5
}
D
6
H
=
{
i
d
,
r
3
}
and
K
=
{
i
d
,
r
2
,
r
4
,
s
,
r
2
s
,
r
4
s
}
K
≅
S
3
D
6
≅
Z
2
×
S
3
S
3
H
K
H
3
H
{
(
1
)
,
(
123
)
,
(
132
)
}
K
2
K
h
k
=
k
h
h
∈
H
k
∈
K
G
H
K
G
H
×
K
G
g
∈
G
g
=
h
k
h
∈
H
k
∈
K
ϕ
:
G
→
H
×
K
ϕ
(
g
)
=
(
h
,
k
)
ϕ
h
k
g
g
=
h
k
=
h
′
k
′
h
−
1
h
′
=
k
(
k
′
)
−
1
H
K
h
=
h
′
k
=
k
′
ϕ
ϕ
g
1
=
h
1
k
1
g
2
=
h
2
k
2
ϕ
(
g
1
g
2
)
=
ϕ
(
h
1
k
1
h
2
k
2
)
=
ϕ
(
h
1
h
2
k
1
k
2
)
=
(
h
1
h
2
,
k
1
k
2
)
=
(
h
1
,
k
1
)
(
h
2
,
k
2
)
=
ϕ
(
g
1
)
ϕ
(
g
2
)
ϕ
Z
6
{
0
,
2
,
4
}
×
{
0
,
3
}
G
H
1
,
H
2
,
…
,
H
n
G
G
=
H
1
H
2
⋯
H
n
=
{
h
1
h
2
⋯
h
n
:
h
i
∈
H
i
}
H
i
∩
⟨
∪
j
i
H
j
⟩
=
{
e
}
h
i
h
j
=
h
j
h
i
h
i
∈
H
i
h
j
∈
H
j
G
H
i
i
=
1
,
2
,
…
,
n
G
∏
i
H
i
(
1
,
2
)
Z
4
×
Z
8
Z
15
Z
≅
n
Z
n
0
Z
C
∗
G
L
2
(
R
)
(
a
b
−
b
a
)
ϕ
:
C
∗
→
G
L
2
(
R
)
ϕ
(
a
+
b
i
)
=
(
a
b
−
b
a
)
U
(
8
)
≅
Z
4
U
(
8
)
(
1
0
0
1
)
,
(
1
0
0
−
1
)
,
(
−
1
0
0
1
)
,
(
−
1
0
0
−
1
)
U
(
5
)
U
(
10
)
U
(
12
)
n
Z
n
Z
n
n
k
↦
cis
(
2
k
π
/
n
)
n
Z
n
Q
Z
Q
G
=
R
∖
{
−
1
}
G
a
∗
b
=
a
+
b
+
a
b
G
(
G
,
∗
)
(
1
0
0
0
1
0
0
0
1
)
(
1
0
0
0
0
1
0
1
0
)
(
0
1
0
1
0
0
0
0
1
)
(
0
0
1
1
0
0
0
1
0
)
(
0
0
1
0
1
0
1
0
0
)
(
0
1
0
0
0
1
1
0
0
)
G
6
8
S
4
D
12
ω
=
cis
(
2
π
/
n
)
n
A
=
(
ω
0
0
ω
−
1
)
and
B
=
(
0
1
1
0
)
D
n
(
±
1
k
0
1
)
D
n
Z
n
Z
4
×
Z
2
(
3
,
4
)
Z
4
×
Z
6
(
6
,
15
,
4
)
Z
30
×
Z
45
×
Z
24
(
5
,
10
,
15
)
Z
25
×
Z
25
×
Z
25
(
8
,
8
,
8
)
Z
10
×
Z
24
×
Z
80
12
5
D
4
Q
∗
2
m
3
n
m
,
n
∈
Z
Z
×
Z
S
3
×
Z
2
D
6
D
2
n
3
3
6
G
20
G
H
K
4
5
h
k
=
k
h
h
∈
H
k
∈
K
G
H
K
G
H
K
G
×
K
≅
H
×
K
G
≅
H
51
52
ϕ
:
G
→
H
ϕ
(
x
)
=
e
H
x
=
e
G
e
G
e
H
G
H
G
≅
H
G
H
a
G
ϕ
:
G
→
H
ϕ
(
a
)
H
G
p
p
Z
p
S
n
A
n
+
2
D
n
S
n
ϕ
:
G
1
→
G
2
ψ
:
G
2
→
G
3
ϕ
−
1
ψ
∘
ϕ
U
(
5
)
≅
Z
4
U
(
p
)
p
S
3
G
ϕ
(
a
+
b
i
)
=
a
−
b
i
C
C
a
+
i
b
↦
a
−
i
b
C
∗
A
↦
B
−
1
A
B
S
L
2
(
R
)
B
G
L
2
(
R
)
G
Aut
(
G
)
G
Aut
(
G
)
S
G
G
Aut
(
Z
6
)
Z
6
Z
6
Aut
(
Z
)
G
H
Aut
(
G
)
≅
Aut
(
H
)
G
g
∈
G
i
g
:
G
→
G
i
g
(
x
)
=
g
x
g
−
1
i
g
G
Inn
(
G
)
i
g
(
x
)
=
g
x
g
−
1
G
Inn
(
G
)
Aut
(
G
)
Q
8
Inn
(
G
)
=
Aut
(
G
)
G
g
∈
G
λ
g
:
G
→
G
ρ
g
:
G
→
G
λ
g
(
x
)
=
g
x
ρ
g
(
x
)
=
x
g
−
1
i
g
=
ρ
g
∘
λ
g
G
g
↦
ρ
g
G
G
H
K
ϕ
:
G
→
H
×
K
ϕ
(
g
)
=
(
h
,
k
)
g
=
h
k
h
∈
H
k
∈
K
ϕ
g
1
=
h
1
k
1
g
2
=
h
2
k
2
ϕ
(
g
1
)
=
ϕ
(
g
2
)
G
H
G
n
H
n
G
≅
G
¯
H
≅
H
¯
G
×
H
≅
G
¯
×
H
¯
G
×
H
H
×
G
n
1
,
…
,
n
k
∏
i
=
1
k
Z
n
i
≅
Z
n
1
⋯
n
k
gcd
(
n
i
,
n
j
)
=
1
i
j
A
×
B
A
B
G
H
1
,
H
2
,
…
,
H
n
G
∏
i
H
i
H
1
H
2
G
1
G
2
H
1
×
H
2
G
1
×
G
2
m
,
n
∈
Z
⟨
m
,
n
⟩
=
⟨
d
⟩
d
=
gcd
(
m
,
n
)
m
,
n
∈
Z
⟨
m
⟩
∩
⟨
n
⟩
=
⟨
l
⟩
l
=
lcm
(
m
,
n
)
2
p
2
p
p
G
2
p
p
a
∈
G
a
1
2
p
2
p
G
2
p
G
Z
2
p
G
G
2
p
G
p
G
p
G
2
p
G
2
P
G
p
y
∈
G
2
y
P
=
P
y
G
2
p
P
=
⟨
z
⟩
p
z
y
2
y
z
=
z
k
y
2
≤
k
<
p
G
2
p
G
G
2
p
P
=
⟨
z
⟩
p
z
y
2
G
{
z
i
y
j
∣
0
≤
i
<
p
,
0
≤
j
<
2
}
G
2
p
P
=
⟨
z
⟩
p
z
y
2
(
z
i
y
j
)
(
z
r
y
s
)
z
m
y
n
m
,
n
2
p
16
Z
n
Z
p
16
1
,
2
,
3
,
5
,
7
,
11
13
4
Z
4
Z
2
×
Z
2
4
4
4
6
Z
3
×
Z
2
Z
6
2
3
D
3
S
3
3
2
p
p
Z
6
D
3
6
6
10
14
n
n
n
n
12
6
2
6
6
2
6
6
6
2
6
2
Z
6
×
Z
2
Z
3
×
Z
2
×
Z
2
3
2
2
Q
Q
±
1
,
±
I
,
±
J
,
±
K
S
8
g
∈
Q
T
g
T
g
(
x
)
=
x
g
T
g
Q
{
1
,
2
,
…
,
8
}
Z
2
×
Z
4
8
Z
2
×
Z
4
Z
2
×
Z
4
U
(
24
)
U
(
24
)
D
10
1
10
180
R
2
72
10
S
10
R
S
G
G
H
G
g
H
=
H
g
g
∈
G
H
G
g
H
=
H
g
g
∈
G
G
G
H
G
g
h
=
h
g
g
∈
G
h
∈
H
g
H
=
H
g
H
S
3
(
1
)
(
12
)
(
123
)
H
=
{
(
123
)
,
(
13
)
}
and
H
(
123
)
=
{
(
123
)
,
(
23
)
}
H
S
3
N
(
1
)
(
123
)
(
132
)
N
N
=
{
(
1
)
,
(
123
)
,
(
132
)
}
(
12
)
N
=
N
(
12
)
=
{
(
12
)
,
(
13
)
,
(
23
)
}
G
N
G
N
G
g
∈
G
g
N
g
−
1
⊂
N
g
∈
G
g
N
g
−
1
=
N
⇒
N
G
g
N
=
N
g
g
∈
G
g
∈
G
n
∈
N
n
′
N
g
n
=
n
′
g
g
n
g
−
1
=
n
′
∈
N
g
N
g
−
1
⊂
N
⇒
g
∈
G
g
N
g
−
1
⊂
N
N
⊂
g
N
g
−
1
n
∈
N
g
−
1
n
g
=
g
−
1
n
(
g
−
1
)
−
1
∈
N
g
−
1
n
g
=
n
′
n
′
∈
N
n
=
g
n
′
g
−
1
g
N
g
−
1
⇒
g
N
g
−
1
=
N
g
∈
G
n
∈
N
n
′
∈
N
g
n
g
−
1
=
n
′
g
n
=
n
′
g
g
N
⊂
N
g
N
g
⊂
g
N
N
G
N
G
G
/
N
(
a
N
)
(
b
N
)
=
a
b
N
G
N
G
N
G
/
N
N
G
N
G
G
/
N
[
G
:
N
]
G
/
N
(
a
N
)
(
b
N
)
=
a
b
N
a
N
=
b
N
c
N
=
d
N
(
a
N
)
(
c
N
)
=
a
c
N
=
b
d
N
=
(
b
N
)
(
d
N
)
a
=
b
n
1
c
=
d
n
2
n
1
n
2
N
a
c
N
=
b
n
1
d
n
2
N
=
b
n
1
d
N
=
b
n
1
N
d
=
b
N
d
=
b
d
N
e
N
=
N
g
−
1
N
g
N
G
/
N
N
G
S
3
N
=
{
(
1
)
,
(
123
)
,
(
132
)
}
N
S
3
N
(
12
)
N
S
3
/
N
N
(
12
)
N
N
N
(
12
)
N
(
12
)
N
(
12
)
N
N
Z
2
S
3
/
N
S
3
N
=
A
3
(
12
)
N
=
{
(
12
)
,
(
13
)
,
(
23
)
}
G
/
N
3
Z
Z
3
Z
Z
0
+
3
Z
=
{
…
,
−
3
,
0
,
3
,
6
,
…
}
1
+
3
Z
=
{
…
,
−
2
,
1
,
4
,
7
,
…
}
2
+
3
Z
=
{
…
,
−
1
,
2
,
5
,
8
,
…
}
Z
/
3
Z
+
0
+
3
Z
1
+
3
Z
2
+
3
Z
0
+
3
Z
0
+
3
Z
1
+
3
Z
2
+
3
Z
1
+
3
Z
1
+
3
Z
2
+
3
Z
0
+
3
Z
2
+
3
Z
2
+
3
Z
0
+
3
Z
1
+
3
Z
n
Z
Z
Z
/
n
Z
n
Z
1
+
n
Z
2
+
n
Z
⋮
(
n
−
1
)
+
n
Z
k
+
n
Z
l
+
n
Z
k
+
l
+
n
Z
D
n
r
s
r
n
=
i
d
s
2
=
i
d
s
r
s
=
r
−
1
r
R
n
D
n
s
r
s
−
1
=
s
r
s
=
r
−
1
∈
R
n
D
n
D
n
/
R
n
Z
2
Z
p
p
A
n
n
≥
5
A
n
3
n
≥
3
3
A
n
3
(
a
b
)
=
(
b
a
)
(
a
b
)
(
a
b
)
=
i
d
(
a
b
)
(
c
d
)
=
(
a
c
b
)
(
a
c
d
)
(
a
b
)
(
a
c
)
=
(
a
c
b
)
N
A
n
n
≥
3
N
3
N
=
A
n
A
n
3
(
i
j
k
)
i
j
{
1
,
2
,
…
,
n
}
k
3
3
(
i
a
j
)
=
(
i
j
a
)
2
(
i
a
b
)
=
(
i
j
b
)
(
i
j
a
)
2
(
j
a
b
)
=
(
i
j
b
)
2
(
i
j
a
)
(
a
b
c
)
=
(
i
j
a
)
2
(
i
j
c
)
(
i
j
b
)
2
(
i
j
a
)
N
A
n
n
≥
3
N
3
(
i
j
a
)
N
[
(
i
j
)
(
a
k
)
]
(
i
j
a
)
2
[
(
i
j
)
(
a
k
)
]
−
1
=
(
i
j
k
)
N
N
3
(
i
j
k
)
1
≤
k
≤
n
3
A
n
N
=
A
n
n
≥
5
N
A
n
3
σ
N
σ
σ
3
σ
σ
=
τ
(
a
1
a
2
⋯
a
r
)
∈
N
r
>
3
σ
σ
=
τ
(
a
1
a
2
a
3
)
(
a
4
a
5
a
6
)
σ
=
τ
(
a
1
a
2
a
3
)
τ
σ
=
τ
(
a
1
a
2
)
(
a
3
a
4
)
τ
σ
3
N
σ
σ
=
τ
(
a
1
a
2
⋯
a
r
)
(
a
1
a
2
a
3
)
σ
(
a
1
a
2
a
3
)
−
1
N
N
σ
−
1
(
a
1
a
2
a
3
)
σ
(
a
1
a
2
a
3
)
−
1
N
σ
−
1
(
a
1
a
2
a
3
)
σ
(
a
1
a
2
a
3
)
−
1
=
σ
−
1
(
a
1
a
2
a
3
)
σ
(
a
1
a
3
a
2
)
=
(
a
1
a
2
⋯
a
r
)
−
1
τ
−
1
(
a
1
a
2
a
3
)
τ
(
a
1
a
2
⋯
a
r
)
(
a
1
a
3
a
2
)
=
(
a
1
a
r
a
r
−
1
⋯
a
2
)
(
a
1
a
2
a
3
)
(
a
1
a
2
⋯
a
r
)
(
a
1
a
3
a
2
)
=
(
a
1
a
3
a
r
)
N
3
N
=
A
n
N
σ
=
τ
(
a
1
a
2
a
3
)
(
a
4
a
5
a
6
)
σ
−
1
(
a
1
a
2
a
4
)
σ
(
a
1
a
2
a
4
)
−
1
∈
N
(
a
1
a
2
a
4
)
σ
(
a
1
a
2
a
4
)
−
1
∈
N
σ
−
1
(
a
1
a
2
a
4
)
σ
(
a
1
a
2
a
4
)
−
1
=
[
τ
(
a
1
a
2
a
3
)
(
a
4
a
5
a
6
)
]
−
1
(
a
1
a
2
a
4
)
τ
(
a
1
a
2
a
3
)
(
a
4
a
5
a
6
)
(
a
1
a
2
a
4
)
−
1
=
(
a
4
a
6
a
5
)
(
a
1
a
3
a
2
)
τ
−
1
(
a
1
a
2
a
4
)
τ
(
a
1
a
2
a
3
)
(
a
4
a
5
a
6
)
(
a
1
a
4
a
2
)
=
(
a
4
a
6
a
5
)
(
a
1
a
3
a
2
)
(
a
1
a
2
a
4
)
(
a
1
a
2
a
3
)
(
a
4
a
5
a
6
)
(
a
1
a
4
a
2
)
=
(
a
1
a
4
a
2
a
6
a
3
)
N
N
σ
=
τ
(
a
1
a
2
a
3
)
τ
σ
∈
N
σ
2
∈
N
σ
2
=
τ
(
a
1
a
2
a
3
)
τ
(
a
1
a
2
a
3
)
=
(
a
1
a
3
a
2
)
N
3
σ
=
τ
(
a
1
a
2
)
(
a
3
a
4
)
τ
2
σ
−
1
(
a
1
a
2
a
3
)
σ
(
a
1
a
2
a
3
)
−
1
N
(
a
1
a
2
a
3
)
σ
(
a
1
a
2
a
3
)
−
1
N
σ
−
1
(
a
1
a
2
a
3
)
σ
(
a
1
a
2
a
3
)
−
1
=
τ
−
1
(
a
1
a
2
)
(
a
3
a
4
)
(
a
1
a
2
a
3
)
τ
(
a
1
a
2
)
(
a
3
a
4
)
(
a
1
a
2
a
3
)
−
1
=
(
a
1
a
3
)
(
a
2
a
4
)
n
≥
5
b
∈
{
1
,
2
,
…
,
n
}
b
a
1
,
a
2
,
a
3
,
a
4
μ
=
(
a
1
a
3
b
)
μ
−
1
(
a
1
a
3
)
(
a
2
a
4
)
μ
(
a
1
a
3
)
(
a
2
a
4
)
∈
N
μ
−
1
(
a
1
a
3
)
(
a
2
a
4
)
μ
(
a
1
a
3
)
(
a
2
a
4
)
=
(
a
1
b
a
3
)
(
a
1
a
3
)
(
a
2
a
4
)
(
a
1
a
3
b
)
(
a
1
a
3
)
(
a
2
a
4
)
=
(
a
1
a
3
b
)
N
3
A
n
n
≥
5
N
A
n
N
3
N
=
A
n
A
n
n
≥
5
A
5
196,833
×
196,833
G
1
,
2
,
3
H
H
=
⟨
(
1
2
)
⟩
H
G
H
G
8
Z
Z
Z
/
8
Z
(
3
+
8
Z
)
+
(
7
+
8
Z
)
G
H
H
G
G
H
G
H
G
/
H
G
=
S
4
H
=
A
4
G
=
A
5
H
=
{
(
1
)
,
(
123
)
,
(
132
)
}
G
=
S
4
H
=
D
4
G
=
Q
8
H
=
{
1
,
−
1
,
I
,
−
I
}
G
=
Z
H
=
5
Z
A
4
(
12
)
A
4
A
4
A
4
(
12
)
A
4
(
12
)
A
4
(
12
)
A
4
A
4
D
4
S
4
D
4
D
4
Q
8
Q
8
T
2
×
2
R
(
a
b
0
c
)
a
b
c
∈
R
a
c
0
U
(
1
x
0
1
)
x
∈
R
U
T
U
U
T
T
/
U
T
G
L
2
(
R
)
G
G
/
H
H
G
H
G
/
H
G
G
G
/
H
a
∈
G
G
a
H
G
/
H
H
G
/
H
G
H
2
G
H
G
S
n
n
≥
3
G
H
k
H
G
g
∈
G
i
g
:
G
→
G
i
g
:
x
↦
g
x
g
−
1
G
i
g
(
H
)
g
G
C
(
g
)
=
{
x
∈
G
:
x
g
=
g
x
}
C
(
g
)
G
g
G
C
(
g
)
G
⟨
g
⟩
G
y
G
x
∈
C
(
g
)
y
x
y
−
1
C
(
g
)
(
y
x
y
−
1
)
g
=
g
(
y
x
y
−
1
)
G
Z
(
G
)
=
{
x
∈
G
:
x
g
=
g
x
for all
g
∈
G
}
S
3
G
L
2
(
R
)
G
G
G
/
Z
(
G
)
G
G
G
′
=
⟨
a
b
a
−
1
b
−
1
⟩
G
′
G
a
b
a
−
1
b
−
1
G
′
G
G
G
′
G
N
G
G
/
N
N
G
g
∈
G
h
∈
G
′
h
=
a
b
a
−
1
b
−
1
g
h
g
−
1
=
g
a
b
a
−
1
b
−
1
g
−
1
=
(
g
a
g
−
1
)
(
g
b
g
−
1
)
(
g
a
−
1
g
−
1
)
(
g
b
−
1
g
−
1
)
=
(
g
a
g
−
1
)
(
g
b
g
−
1
)
(
g
a
g
−
1
)
−
1
(
g
b
g
−
1
)
−
1
h
=
h
1
⋯
h
n
h
i
=
a
i
b
i
a
i
−
1
b
i
−
1
g
h
g
−
1
g
h
g
−
1
=
g
h
1
⋯
h
n
g
−
1
=
(
g
h
1
g
−
1
)
(
g
h
2
g
−
1
)
⋯
(
g
h
n
g
−
1
)
D
8
8
1
3
4
1
3
1
3
2
1
3
2
1
3
1
3
A
4
1
3
3
1
3
0
2
A
5
A
5
A
5
A
5
4
4
4
4
Z
2
×
Z
2
4
A
5
8
4
4
8
4
8
3
2
n
n
D
n
3
≤
n
≤
100
D
470448
D
470448
(
G
,
⋅
)
(
H
,
∘
)
ϕ
:
G
→
H
ϕ
(
g
1
⋅
g
2
)
=
ϕ
(
g
1
)
∘
ϕ
(
g
2
)
g
1
,
g
2
∈
G
ϕ
H
ϕ
S
n
Z
2
S
n
Z
2
even
odd
even
even
odd
odd
odd
even
G
g
∈
G
ϕ
:
Z
→
G
ϕ
(
n
)
=
g
n
ϕ
ϕ
(
m
+
n
)
=
g
m
+
n
=
g
m
g
n
=
ϕ
(
m
)
ϕ
(
n
)
Z
G
g
G
=
G
L
2
(
R
)
A
=
(
a
b
c
d
)
G
det
(
A
)
=
a
d
−
b
c
0
A
B
G
det
(
A
B
)
=
det
(
A
)
det
(
B
)
ϕ
:
G
L
2
(
R
)
→
R
∗
A
↦
det
(
A
)
T
z
|
z
|
=
1
ϕ
R
T
ϕ
:
θ
↦
cos
θ
+
i
sin
θ
ϕ
(
α
+
β
)
=
cos
(
α
+
β
)
+
i
sin
(
α
+
β
)
=
(
cos
α
cos
β
−
sin
α
sin
β
)
+
i
(
sin
α
cos
β
+
cos
α
sin
β
)
=
(
cos
α
+
i
sin
α
)
(
cos
β
+
i
sin
β
)
=
ϕ
(
α
)
ϕ
(
β
)
ϕ
:
G
1
→
G
2
e
G
1
ϕ
(
e
)
G
2
g
∈
G
1
ϕ
(
g
−
1
)
=
[
ϕ
(
g
)
]
−
1
H
1
G
1
ϕ
(
H
1
)
G
2
H
2
G
2
ϕ
−
1
(
H
2
)
=
{
g
∈
G
1
:
ϕ
(
g
)
∈
H
2
}
G
1
H
2
G
2
ϕ
−
1
(
H
2
)
G
1
e
e
′
G
1
G
2
e
′
ϕ
(
e
)
=
ϕ
(
e
)
=
ϕ
(
e
e
)
=
ϕ
(
e
)
ϕ
(
e
)
ϕ
(
e
)
=
e
′
ϕ
(
g
−
1
)
ϕ
(
g
)
=
ϕ
(
g
−
1
g
)
=
ϕ
(
e
)
=
e
′
ϕ
(
H
1
)
G
2
ϕ
(
H
1
)
H
1
G
1
x
y
ϕ
(
H
1
)
a
,
b
∈
H
1
ϕ
(
a
)
=
x
ϕ
(
b
)
=
y
x
y
−
1
=
ϕ
(
a
)
[
ϕ
(
b
)
]
−
1
=
ϕ
(
a
b
−
1
)
∈
ϕ
(
H
1
)
ϕ
(
H
1
)
G
2
H
2
G
2
H
1
ϕ
−
1
(
H
2
)
H
1
g
∈
G
1
ϕ
(
g
)
∈
H
2
H
1
ϕ
(
e
)
=
e
′
a
b
H
1
ϕ
(
a
b
−
1
)
=
ϕ
(
a
)
[
ϕ
(
b
)
]
−
1
H
2
H
2
G
2
a
b
−
1
∈
H
1
H
1
G
1
H
2
G
2
g
−
1
h
g
∈
H
1
h
∈
H
1
g
∈
G
1
ϕ
(
g
−
1
h
g
)
=
[
ϕ
(
g
)
]
−
1
ϕ
(
h
)
ϕ
(
g
)
∈
H
2
H
2
G
2
g
−
1
h
g
∈
H
1
ϕ
:
G
→
H
e
H
ϕ
−
1
(
{
e
}
)
G
ϕ
ker
ϕ
ϕ
G
H
ϕ
:
G
→
H
ϕ
G
ϕ
:
G
L
2
(
R
)
→
R
∗
A
↦
det
(
A
)
1
R
∗
2
×
2
ker
ϕ
=
S
L
2
(
R
)
ϕ
:
R
→
C
∗
ϕ
(
θ
)
=
cos
θ
+
i
sin
θ
{
2
π
n
:
n
∈
Z
}
ker
ϕ
≅
Z
ϕ
Z
7
Z
12
ϕ
Z
7
{
0
}
Z
7
Z
7
Z
12
Z
12
7
Z
7
Z
12
G
g
∈
G
ϕ
Z
G
ϕ
(
n
)
=
g
n
g
{
0
}
ϕ
Z
G
g
g
n
ϕ
n
Z
ϕ
:
G
→
H
G
ker
ϕ
G
H
G
ϕ
:
G
→
G
/
H
ϕ
(
g
)
=
g
H
ϕ
(
g
1
g
2
)
=
g
1
g
2
H
=
g
1
H
g
2
H
=
ϕ
(
g
1
)
ϕ
(
g
2
)
H
ψ
:
G
→
H
K
=
ker
ψ
K
G
ϕ
:
G
→
G
/
K
η
:
G
/
K
→
ψ
(
G
)
ψ
=
η
ϕ
K
G
η
:
G
/
K
→
ψ
(
G
)
η
(
g
K
)
=
ψ
(
g
)
η
g
1
K
=
g
2
K
k
∈
K
g
1
k
=
g
2
η
(
g
1
K
)
=
ψ
(
g
1
)
=
ψ
(
g
1
)
ψ
(
k
)
=
ψ
(
g
1
k
)
=
ψ
(
g
2
)
=
η
(
g
2
K
)
η
η
:
G
/
K
→
ψ
(
G
)
ψ
=
η
ϕ
η
η
(
g
1
K
g
2
K
)
=
η
(
g
1
g
2
K
)
=
ψ
(
g
1
g
2
)
=
ψ
(
g
1
)
ψ
(
g
2
)
=
η
(
g
1
K
)
η
(
g
2
K
)
η
ψ
(
G
)
η
η
(
g
1
K
)
=
η
(
g
2
K
)
ψ
(
g
1
)
=
ψ
(
g
2
)
ψ
(
g
1
−
1
g
2
)
=
e
g
1
−
1
g
2
ψ
g
1
−
1
g
2
K
=
K
g
1
K
=
g
2
K
ψ
=
η
ϕ
G
g
ϕ
:
Z
→
G
n
↦
g
n
ϕ
(
m
+
n
)
=
g
m
+
n
=
g
m
g
n
=
ϕ
(
m
)
ϕ
(
n
)
ϕ
|
g
|
=
m
g
m
=
e
ker
ϕ
=
m
Z
Z
/
ker
ϕ
=
Z
/
m
Z
≅
G
g
ker
ϕ
=
0
ϕ
G
Z
Z
Z
n
H
G
G
N
G
H
N
G
H
∩
N
H
H
/
H
∩
N
≅
H
N
/
N
H
N
=
{
h
n
:
h
∈
H
,
n
∈
N
}
G
h
1
n
1
,
h
2
n
2
∈
H
N
N
(
h
2
)
−
1
n
1
h
2
∈
N
(
h
1
n
1
)
(
h
2
n
2
)
=
h
1
h
2
(
(
h
2
)
−
1
n
1
h
2
)
n
2
H
N
h
n
∈
H
N
H
N
(
h
n
)
−
1
=
n
−
1
h
−
1
=
h
−
1
(
h
n
−
1
h
−
1
)
H
∩
N
H
h
∈
H
n
∈
H
∩
N
h
−
1
n
h
∈
H
H
h
−
1
n
h
∈
N
N
G
h
−
1
n
h
∈
H
∩
N
ϕ
H
H
N
/
N
h
↦
h
N
ϕ
h
n
N
=
h
N
h
H
ϕ
ϕ
(
h
h
′
)
=
h
h
′
N
=
h
N
h
′
N
=
ϕ
(
h
)
ϕ
(
h
′
)
ϕ
H
/
ker
ϕ
H
N
/
N
=
ϕ
(
H
)
≅
H
/
ker
ϕ
ker
ϕ
=
{
h
∈
H
:
h
∈
N
}
=
H
∩
N
H
N
/
N
=
ϕ
(
H
)
≅
H
/
H
∩
N
N
G
H
↦
H
/
N
H
N
G
/
N
G
N
G
/
N
H
G
N
N
H
H
/
N
a
N
b
N
H
/
N
(
a
N
)
(
b
−
1
N
)
=
a
b
−
1
N
∈
H
/
N
H
/
N
G
/
N
S
G
/
N
N
H
=
{
g
∈
G
:
g
N
∈
S
}
h
1
,
h
2
∈
H
(
h
1
N
)
(
h
2
N
)
=
h
1
h
2
N
∈
S
h
1
−
1
N
∈
S
H
G
H
N
S
=
H
/
N
H
↦
H
/
N
H
1
H
2
G
N
H
1
/
N
=
H
2
/
N
h
1
∈
H
1
h
1
N
∈
H
1
/
N
h
1
N
=
h
2
N
⊂
H
2
h
2
H
2
N
H
2
h
1
∈
H
2
H
1
⊂
H
2
H
2
⊂
H
1
H
1
=
H
2
H
↦
H
/
N
H
G
N
H
G
/
N
→
G
/
H
g
N
↦
g
H
H
/
N
H
/
N
G
/
N
H
/
N
G
/
N
G
→
G
/
N
→
G
/
N
H
/
N
H
H
G
G
N
H
G
N
⊂
H
G
/
H
≅
G
/
N
H
/
N
Z
/
m
Z
≅
(
Z
/
m
n
Z
)
/
(
m
Z
/
m
n
Z
)
|
Z
/
m
n
Z
|
=
m
n
|
Z
/
m
Z
|
=
m
|
m
Z
/
m
n
Z
|
=
n
ϕ
:
Z
10
→
Z
10
ϕ
(
x
)
=
x
+
x
ϕ
ϕ
ϕ
det
(
A
B
)
=
det
(
A
)
det
(
B
)
A
,
B
∈
G
L
2
(
R
)
G
L
2
(
R
)
R
∗
ϕ
:
R
∗
→
G
L
2
(
R
)
ϕ
(
a
)
=
(
1
0
0
a
)
ϕ
:
R
→
G
L
2
(
R
)
ϕ
(
a
)
=
(
1
0
a
1
)
ϕ
:
G
L
2
(
R
)
→
R
ϕ
(
(
a
b
c
d
)
)
=
a
+
d
ϕ
:
G
L
2
(
R
)
→
R
∗
ϕ
(
(
a
b
c
d
)
)
=
a
d
−
b
c
ϕ
:
M
2
(
R
)
→
R
ϕ
(
(
a
b
c
d
)
)
=
b
M
2
(
R
)
2
×
2
R
{
1
}
A
m
×
n
x
↦
A
x
ϕ
:
R
n
→
R
m
ϕ
:
Z
→
Z
ϕ
(
n
)
=
7
n
ϕ
ϕ
ϕ
(
m
+
n
)
=
7
(
m
+
n
)
=
7
m
+
7
n
=
ϕ
(
m
)
+
ϕ
(
n
)
ϕ
Z
24
Z
18
ϕ
:
Z
24
→
Z
18
ϕ
Z
24
ϕ
Z
18
Z
Z
12
Z
24
H
=
⟨
4
⟩
N
=
⟨
6
⟩
H
N
H
+
N
H
∩
N
H
N
/
N
H
/
(
H
∩
N
)
H
N
/
N
H
/
(
H
∩
N
)
G
n
∈
N
ϕ
:
G
→
G
g
↦
g
n
ϕ
:
G
→
H
G
ϕ
(
G
)
a
,
b
∈
G
ϕ
(
a
)
ϕ
(
b
)
=
ϕ
(
a
b
)
=
ϕ
(
b
a
)
=
ϕ
(
b
)
ϕ
(
a
)
ϕ
:
G
→
H
G
ϕ
(
G
)
G
H
k
H
G
Q
/
Z
≅
Q
G
N
G
H
G
/
N
ϕ
−
1
(
H
)
G
|
H
|
⋅
|
N
|
ϕ
:
G
→
G
/
N
G
1
G
2
H
1
H
2
G
1
G
2
ϕ
:
G
1
→
G
2
ϕ
ϕ
¯
:
(
G
1
/
H
1
)
→
(
G
2
/
H
2
)
ϕ
(
H
1
)
⊂
H
2
H
K
G
H
∩
K
=
{
e
}
G
G
/
H
×
G
/
K
ϕ
:
G
1
→
G
2
H
1
G
1
ϕ
(
H
1
)
=
H
2
G
1
/
H
1
≅
G
2
/
H
2
ϕ
:
G
→
H
ϕ
ϕ
−
1
(
e
)
=
{
e
}
ϕ
:
G
→
H
∼
G
a
∼
b
ϕ
(
a
)
=
ϕ
(
b
)
a
,
b
∈
G
Aut
(
G
)
G
G
G
Aut
(
G
)
≤
S
G
G
i
g
:
G
→
G
i
g
(
x
)
=
g
x
g
−
1
g
∈
G
i
g
∈
Aut
(
G
)
Inn
(
G
)
Inn
(
G
)
Aut
(
G
)
G
G
i
g
G
G
→
Aut
(
G
)
g
↦
i
g
Inn
(
G
)
Z
(
G
)
G
/
Z
(
G
)
≅
Inn
(
G
)
Aut
(
S
3
)
Inn
(
S
3
)
D
4
ϕ
:
Z
→
Z
Aut
(
Z
)
Z
8
Aut
(
Z
8
)
≅
U
(
8
)
k
∈
Z
n
ϕ
k
:
Z
n
→
Z
n
a
↦
k
a
ϕ
k
ϕ
k
k
Z
n
Z
n
ϕ
k
k
Z
n
ψ
:
U
(
n
)
→
Aut
(
Z
n
)
ψ
:
k
↦
ϕ
k
12
20
G
=
{
a
i
|
a
12
=
e
}
H
=
{
x
i
|
x
20
=
e
}
G
ϕ
:
G
→
H
,
ϕ
(
a
)
=
x
5
⇒
ϕ
(
a
i
)
=
ϕ
(
a
)
i
=
(
x
5
)
i
=
x
5
i
4
3
12
4
20
D
20
20
5
D
20
D
5
5
72
7
4
G
×
H
G
H
G
×
H
x
↦
x
12
12
G
H
G
×
H
S
7
(
1
,
2
,
3
)
(
4
,
5
,
6
,
7
)
S
12
(
1
,
2
,
3
)
(
4
,
5
,
6
)
(
7
,
8
,
9
)
(
10
,
11
,
12
)
(
1
,
10
,
7
,
4
)
(
2
,
11
,
8
,
5
)
(
3
,
12
,
9
,
6
)
S
n
Z
2
S
n
S
6
Z
2
2
2
S
6
D
20
D
20
D
20
D
20
D
20
20
T
:
R
n
→
R
m
x
y
R
n
α
∈
R
T
(
x
+
y
)
=
T
(
x
)
+
T
(
y
)
T
(
α
y
)
=
α
T
(
y
)
m
×
n
R
R
n
R
m
x
=
(
x
1
,
…
,
x
n
)
t
y
=
(
y
1
,
…
,
y
n
)
t
R
n
m
×
n
A
=
(
a
11
a
12
⋯
a
1
n
a
21
a
22
⋯
a
2
n
⋮
⋮
⋱
⋮
a
m
1
a
m
2
⋯
a
m
n
)
R
m
α
A
(
x
+
y
)
=
A
x
+
A
y
and
α
A
x
=
A
(
α
x
)
x
=
(
x
1
x
2
⋮
x
n
)
A
(
a
i
j
)
T
:
R
n
→
R
m
A
T
T
e
1
=
(
1
,
0
,
…
,
0
)
t
e
2
=
(
0
,
1
,
…
,
0
)
t
⋮
e
n
=
(
0
,
0
,
…
,
1
)
t
x
=
(
x
1
,
…
,
x
n
)
t
x
1
e
1
+
x
2
e
2
+
⋯
+
x
n
e
n
T
(
e
1
)
=
(
a
11
,
a
21
,
…
,
a
m
1
)
t
,
T
(
e
2
)
=
(
a
12
,
a
22
,
…
,
a
m
2
)
t
,
⋮
T
(
e
n
)
=
(
a
1
n
,
a
2
n
,
…
,
a
m
n
)
t
T
(
x
)
=
T
(
x
1
e
1
+
x
2
e
2
+
⋯
+
x
n
e
n
)
=
x
1
T
(
e
1
)
+
x
2
T
(
e
2
)
+
⋯
+
x
n
T
(
e
n
)
=
(
∑
k
=
1
n
a
1
k
x
k
,
…
,
∑
k
=
1
n
a
m
k
x
k
)
t
=
A
x
T
:
R
2
→
R
2
T
(
x
1
,
x
2
)
=
(
2
x
1
+
5
x
2
,
−
4
x
1
+
3
x
2
)
T
T
e
1
=
(
2
,
−
4
)
t
T
e
2
=
(
5
,
3
)
t
T
A
=
(
2
5
−
4
3
)
n
×
n
A
A
−
1
A
A
−
1
=
A
−
1
A
=
I
I
=
(
1
0
⋯
0
0
1
⋯
0
⋮
⋮
⋱
⋮
0
0
⋯
1
)
n
×
n
A
A
A
(
2
1
5
3
)
A
A
−
1
=
(
3
−
1
−
5
2
)
A
−
1
det
(
A
)
=
2
⋅
3
−
5
⋅
1
=
1
A
B
n
×
n
det
(
A
B
)
=
(
det
A
)
(
det
B
)
A
det
(
A
−
1
)
=
1
/
det
A
A
=
(
a
i
j
)
A
t
=
(
a
j
i
)
det
(
A
t
)
=
det
A
T
n
×
n
A
T
|
det
A
|
R
2
T
|
det
A
|
2
×
2
n
×
n
G
L
n
(
R
)
det
(
A
)
=
1
det
(
B
)
=
1
det
(
A
B
)
=
det
(
A
)
det
(
B
)
=
1
det
(
A
−
1
)
=
1
/
det
A
=
1
S
L
n
(
R
)
2
×
2
A
=
(
a
b
c
d
)
A
a
d
−
b
c
G
L
2
(
R
)
a
d
−
b
c
0
A
A
−
1
=
1
a
d
−
b
c
(
d
−
b
−
c
a
)
A
S
L
2
(
R
)
A
−
1
=
(
d
−
b
−
c
a
)
S
L
2
(
R
)
A
=
(
1
1
0
1
)
S
L
2
(
R
)
x
=
(
1
,
0
)
t
y
=
(
0
,
1
)
t
A
(
1
,
0
)
t
(
1
,
1
)
t
A
x
=
(
1
,
0
)
t
A
y
=
(
1
,
1
)
t
S
L
2
(
R
)
O
(
n
)
G
L
n
(
R
)
A
A
−
1
=
A
t
O
(
n
)
n
×
n
O
(
n
)
G
L
n
(
R
)
(
3
/
5
−
4
/
5
4
/
5
3
/
5
)
,
(
1
/
2
−
3
/
2
3
/
2
1
/
2
)
,
(
−
1
/
2
0
1
/
2
1
/
6
−
2
/
6
1
/
6
1
/
3
1
/
3
1
/
3
)
O
(
n
)
x
=
(
x
1
,
…
,
x
n
)
t
y
=
(
y
1
,
…
,
y
n
)
t
⟨
x
,
y
⟩
=
x
t
y
=
(
x
1
,
x
2
,
…
,
x
n
)
(
y
1
y
2
⋮
y
n
)
=
x
1
y
1
+
⋯
+
x
n
y
n
x
=
(
x
1
,
…
,
x
n
)
t
x
‖
x
‖
=
⟨
x
,
x
⟩
=
x
1
2
+
⋯
+
x
n
2
x
y
‖
x
−
y
‖
x
y
w
R
n
α
∈
R
⟨
x
,
y
⟩
=
⟨
y
,
x
⟩
⟨
x
,
y
+
w
⟩
=
⟨
x
,
y
⟩
+
⟨
x
,
w
⟩
⟨
α
x
,
y
⟩
=
⟨
x
,
α
y
⟩
=
α
⟨
x
,
y
⟩
⟨
x
,
x
⟩
≥
0
x
=
0
⟨
x
,
y
⟩
=
0
x
R
n
y
=
0
x
=
(
3
,
4
)
t
3
2
+
4
2
=
5
A
=
(
3
/
5
−
4
/
5
4
/
5
3
/
5
)
A
x
=
(
−
7
/
5
,
24
/
5
)
t
det
(
A
A
t
)
=
det
(
I
)
=
1
det
(
A
)
=
det
(
A
t
)
1
−
1
a
j
=
(
a
1
j
a
2
j
⋮
a
n
j
)
A
=
(
a
i
j
)
A
A
t
=
I
⟨
a
r
,
a
s
⟩
=
δ
r
s
δ
r
s
=
{
1
r
=
s
0
r
s
n
×
n
A
A
−
1
=
A
t
A
‖
A
x
−
A
y
‖
=
‖
x
−
y
‖
‖
A
x
‖
=
‖
x
‖
⟨
A
x
,
A
y
⟩
=
⟨
x
,
y
⟩
A
n
×
n
A
A
−
1
=
A
t
x
y
⟨
A
x
,
A
y
⟩
=
⟨
x
,
y
⟩
x
y
‖
A
x
−
A
y
‖
=
‖
x
−
y
‖
x
‖
A
x
‖
=
‖
x
‖
(
2
)
⇒
(
3
)
⟨
A
x
,
A
y
⟩
=
(
A
x
)
t
A
y
=
x
t
A
t
A
y
=
x
t
y
=
⟨
x
,
y
⟩
(
3
)
⇒
(
2
)
⟨
x
,
x
⟩
=
⟨
A
x
,
A
x
⟩
=
x
t
A
t
A
x
=
⟨
x
,
A
t
A
x
⟩
⟨
x
,
(
A
t
A
−
I
)
x
⟩
=
0
x
A
t
A
−
I
=
0
A
−
1
=
A
t
(
3
)
⇒
(
4
)
A
A
‖
A
x
−
A
y
‖
2
=
‖
A
(
x
−
y
)
‖
2
=
⟨
A
(
x
−
y
)
,
A
(
x
−
y
)
⟩
=
⟨
x
−
y
,
x
−
y
⟩
=
‖
x
−
y
‖
2
(
4
)
⇒
(
5
)
A
A
y
=
0
‖
A
x
‖
=
‖
A
x
−
A
y
‖
=
‖
x
−
y
‖
=
‖
x
‖
(
5
)
⇒
(
3
)
⟨
x
,
y
⟩
=
1
2
[
‖
x
+
y
‖
2
−
‖
x
‖
2
−
‖
y
‖
2
]
⟨
A
x
,
A
y
⟩
=
1
2
[
‖
A
x
+
A
y
‖
2
−
‖
A
x
‖
2
−
‖
A
y
‖
2
]
=
1
2
[
‖
A
(
x
+
y
)
‖
2
−
‖
A
x
‖
2
−
‖
A
y
‖
2
]
=
1
2
[
‖
x
+
y
‖
2
−
‖
x
‖
2
−
‖
y
‖
2
]
=
⟨
x
,
y
⟩
O
(
2
)
R
2
R
2
A
∈
O
(
2
)
e
1
=
(
1
,
0
)
t
e
2
=
(
0
,
1
)
t
A
e
1
=
(
a
,
b
)
t
a
2
+
b
2
=
1
A
O
(
2
)
A
e
2
=
±
(
−
b
,
a
)
t
A
e
2
=
(
−
b
,
a
)
t
A
=
(
a
−
b
b
a
)
=
(
cos
θ
−
sin
θ
sin
θ
cos
θ
)
,
0
≤
θ
<
2
π
A
R
2
θ
A
e
2
=
(
b
,
−
a
)
t
B
=
(
a
b
b
−
a
)
=
(
cos
θ
sin
θ
sin
θ
−
cos
θ
)
.
det
B
=
−
1
B
2
=
(
1
0
0
1
)
.
C
=
(
1
0
0
−
1
)
,
B
=
A
C
ℓ
S
O
(
n
)
O
(
n
)
S
L
n
(
R
)
O
(
n
)
E
(
n
)
(
A
,
x
)
A
O
(
n
)
x
R
n
(
A
,
x
)
(
B
,
y
)
=
(
A
B
,
A
y
+
x
)
(
I
,
0
)
(
A
,
x
)
(
A
−
1
,
−
A
−
1
x
)
E
(
n
)
R
2
R
n
f
R
n
R
n
f
‖
f
(
x
)
−
f
(
y
)
‖
=
‖
x
−
y
‖
x
,
y
∈
R
n
f
O
(
n
)
R
n
O
(
n
)
R
n
x
T
y
(
x
)
=
x
+
y
T
O
(
n
)
R
2
R
2
R
n
f
R
2
f
O
(
2
)
f
R
2
f
f
(
0
)
=
0
‖
f
(
x
)
‖
=
‖
x
‖
‖
x
‖
2
−
2
⟨
f
(
x
)
,
f
(
y
)
⟩
+
‖
y
‖
2
=
‖
f
(
x
)
‖
2
−
2
⟨
f
(
x
)
,
f
(
y
)
⟩
+
‖
f
(
y
)
‖
2
=
⟨
f
(
x
)
−
f
(
y
)
,
f
(
x
)
−
f
(
y
)
⟩
=
‖
f
(
x
)
−
f
(
y
)
‖
2
=
‖
x
−
y
‖
2
=
⟨
x
−
y
,
x
−
y
⟩
=
‖
x
‖
2
−
2
⟨
x
,
y
⟩
+
‖
y
‖
2
⟨
f
(
x
)
,
f
(
y
)
⟩
=
⟨
x
,
y
⟩
e
1
e
2
(
1
,
0
)
t
(
0
,
1
)
t
x
=
(
x
1
,
x
2
)
=
x
1
e
1
+
x
2
e
2
f
(
x
)
=
⟨
f
(
x
)
,
f
(
e
1
)
⟩
f
(
e
1
)
+
⟨
f
(
x
)
,
f
(
e
2
)
⟩
f
(
e
2
)
=
x
1
f
(
e
1
)
+
x
2
f
(
e
2
)
f
f
T
x
f
x
R
2
T
x
f
(
y
)
=
A
y
A
∈
O
(
2
)
f
(
y
)
=
A
y
+
x
f
(
y
)
=
A
y
+
x
1
g
(
y
)
=
B
y
+
x
2
f
(
g
(
y
)
)
=
f
(
B
y
+
x
2
)
=
A
B
y
+
A
x
2
+
x
1
R
2
E
(
2
)
R
2
E
(
2
)
R
n
R
n
X
⊂
R
n
X
R
n
X
R
1
Z
2
R
2
O
(
2
)
R
2
Z
n
D
n
G
E
(
2
)
G
R
2
O
(
2
)
O
(
2
)
R
θ
=
(
cos
θ
−
sin
θ
sin
θ
cos
θ
)
T
ϕ
=
(
cos
ϕ
−
sin
ϕ
sin
ϕ
cos
ϕ
)
(
1
0
0
−
1
)
=
(
cos
ϕ
sin
ϕ
sin
ϕ
−
cos
ϕ
)
det
(
R
θ
)
=
1
det
(
T
ϕ
)
=
−
1
T
ϕ
2
=
I
G
G
−
1
G
G
G
θ
0
R
θ
0
R
θ
0
G
n
θ
1
n
θ
0
(
n
+
1
)
θ
0
(
n
+
1
)
θ
0
−
θ
1
θ
0
θ
0
G
T
ϕ
:
G
→
{
−
1
,
1
}
A
↦
det
(
A
)
|
G
/
ker
ϕ
|
=
2
G
n
|
G
|
=
2
n
G
R
θ
,
…
,
R
θ
n
−
1
,
T
R
θ
,
…
,
T
R
θ
n
−
1
T
R
θ
T
=
R
θ
−
1
G
D
n
R
3
R
2
R
3
x
y
R
2
x
y
m
x
+
n
y
m
n
x
y
(
1
,
1
)
t
(
2
,
0
)
t
(
−
1
,
1
)
t
(
−
1
,
−
1
)
t
{
x
1
,
x
2
}
{
y
1
,
y
2
}
y
1
=
α
1
x
1
+
α
2
x
2
y
2
=
β
1
x
1
+
β
2
x
2
α
1
α
2
β
1
β
2
U
=
(
α
1
α
2
β
1
β
2
)
x
1
x
2
y
1
y
2
U
−
1
U
−
1
(
y
1
y
2
)
=
(
x
1
x
2
)
U
U
−
1
U
U
−
1
U
U
−
1
=
I
det
(
U
U
−
1
)
=
det
(
U
)
det
(
U
−
1
)
=
1
;
det
(
U
)
=
±
1
±
1
(
3
1
5
2
)
R
2
E
(
2
)
R
2
R
3
G
⊂
E
(
2
)
{
(
I
,
t
)
:
t
∈
L
}
L
R
2
Z
×
Z
G
G
0
=
{
A
:
(
A
,
b
)
∈
G
for some
b
}
G
0
O
(
2
)
x
L
G
H
G
0
(
A
,
y
)
G
(
A
,
y
)
(
I
,
x
)
(
A
,
y
)
−
1
=
(
A
,
A
x
+
y
)
(
A
−
1
,
−
A
−
1
y
)
=
(
A
A
−
1
,
−
A
A
−
1
y
+
A
x
+
y
)
=
(
I
,
A
x
)
;
(
I
,
A
x
)
G
A
x
L
G
0
G
T
G
G
/
T
≅
G
0
Z
n
D
n
n
=
1
,
2
,
3
,
4
,
6
R
2
Z
1
Z
2
Z
3
Z
4
Z
6
D
1
D
1
D
1
D
2
D
2
D
2
D
2
D
3
D
4
D
6
n
R
3
R
4
R
5
R
3
2
×
2
R
n
R
2
⟨
x
,
y
⟩
=
1
2
[
‖
x
+
y
‖
2
−
‖
x
‖
2
−
‖
y
‖
2
]
1
2
[
‖
x
+
y
‖
2
+
‖
x
‖
2
−
‖
y
‖
2
]
=
1
2
[
⟨
x
+
y
,
x
+
y
⟩
−
‖
x
‖
2
−
‖
y
‖
2
]
=
1
2
[
‖
x
‖
2
+
2
⟨
x
,
y
⟩
+
‖
y
‖
2
−
‖
x
‖
2
−
‖
y
‖
2
]
=
⟨
x
,
y
⟩
O
(
n
)
S
O
(
n
)
(
1
/
2
−
1
/
2
1
/
2
1
/
2
)
(
1
/
5
2
/
5
−
2
/
5
1
/
5
)
(
4
/
5
0
3
/
5
−
3
/
5
0
4
/
5
0
−
1
0
)
(
1
/
3
2
/
3
−
2
/
3
−
2
/
3
2
/
3
1
/
3
−
2
/
3
1
/
3
2
/
3
)
S
O
(
2
)
O
(
3
)
x
y
w
R
n
α
∈
R
⟨
x
,
y
⟩
=
⟨
y
,
x
⟩
⟨
x
,
y
+
w
⟩
=
⟨
x
,
y
⟩
+
⟨
x
,
w
⟩
⟨
α
x
,
y
⟩
=
⟨
x
,
α
y
⟩
=
α
⟨
x
,
y
⟩
⟨
x
,
x
⟩
≥
0
x
=
0
⟨
x
,
y
⟩
=
0
x
R
n
y
=
0
⟨
x
,
y
⟩
=
⟨
y
,
x
⟩
E
(
n
)
=
{
(
A
,
x
)
:
A
∈
O
(
n
)
and
x
∈
R
n
}
{
(
2
,
1
)
,
(
1
,
1
)
}
{
(
12
,
5
)
,
(
7
,
3
)
}
(
5
2
2
1
)
G
E
(
2
)
T
G
G
G
/
T
A
∈
S
L
2
(
R
)
x
y
R
2
A
x
A
y
S
O
(
n
)
O
(
n
)
det
:
O
(
n
)
→
R
∗
S
O
(
n
)
f
R
n
E
(
2
)
(
A
,
x
)
x
0
O
(
n
)
x
=
(
x
1
,
x
2
)
R
2
x
1
2
+
x
2
2
=
1
A
∈
O
(
2
)
A
x
G
H
N
G
N
H
H
∩
N
=
{
i
d
}
H
N
=
G
S
3
A
3
H
=
{
(
1
)
,
(
12
)
}
Q
8
E
(
2
)
O
(
2
)
H
H
R
2
p
6
m
n
Z
n
G
=
H
n
⊃
H
n
−
1
⊃
⋯
⊃
H
1
⊃
H
0
=
{
e
}
H
i
H
i
+
1
H
i
+
1
/
H
i
G
Z
p
p
Z
m
n
≅
Z
m
×
Z
n
gcd
(
m
,
n
)
=
1
Z
p
1
α
1
×
⋯
×
Z
p
n
α
n
p
k
G
{
g
i
}
G
i
I
G
g
i
G
g
i
G
G
G
{
g
i
:
i
∈
I
}
g
i
G
{
g
i
:
i
∈
I
}
G
G
S
3
(
12
)
(
123
)
Z
×
Z
n
{
(
1
,
0
)
,
(
0
,
1
)
}
Q
Q
p
1
/
q
1
,
…
,
p
n
/
q
n
p
i
/
q
i
p
q
1
,
…
,
q
n
1
/
p
Q
p
1
/
q
1
,
…
,
p
n
/
q
n
p
p
i
/
q
i
+
p
j
/
q
j
=
(
p
i
q
j
+
p
j
q
i
)
/
(
q
i
q
j
)
H
G
{
g
i
∈
G
:
i
∈
I
}
h
∈
H
h
=
g
i
1
α
1
⋯
g
i
n
α
n
g
i
k
K
g
i
1
α
1
⋯
g
i
n
α
n
g
i
k
K
H
K
G
K
=
H
H
g
i
K
g
i
0
=
1
K
g
=
g
i
1
k
1
⋯
g
i
n
k
n
K
K
g
−
1
=
(
g
i
1
k
1
⋯
g
i
n
k
n
)
−
1
=
(
g
i
n
−
k
n
⋯
g
i
1
−
k
1
)
g
i
g
i
a
−
3
b
5
a
7
a
4
b
5
p
G
p
p
G
p
Z
2
×
Z
2
Z
4
2
Z
27
3
p
G
Z
p
1
α
1
×
Z
p
2
α
2
×
⋯
×
Z
p
n
α
n
p
i
540
=
2
2
⋅
3
3
⋅
5
Z
2
×
Z
2
×
Z
3
×
Z
3
×
Z
3
×
Z
5
Z
2
×
Z
2
×
Z
3
×
Z
9
×
Z
5
Z
2
×
Z
2
×
Z
27
×
Z
5
Z
4
×
Z
3
×
Z
3
×
Z
3
×
Z
5
Z
4
×
Z
3
×
Z
9
×
Z
5
Z
4
×
Z
27
×
Z
5
G
n
p
n
G
p
n
=
1
k
k
<
n
p
n
G
G
=
⟨
a
⟩
a
G
p
n
p
=
n
G
p
−
1
p
G
H
1
<
|
H
|
<
n
p
∣
|
H
|
H
p
p
H
G
H
G
|
G
|
=
|
H
|
⋅
|
G
/
H
|
p
|
G
/
H
|
|
G
/
H
|
<
|
G
|
=
n
G
/
H
a
H
p
H
=
(
a
H
)
p
=
a
p
H
a
p
∈
H
a
∉
H
|
H
|
=
r
p
r
s
t
s
p
+
t
r
=
1
a
p
r
(
a
p
)
r
=
(
a
r
)
p
=
1
a
r
p
a
r
1
a
r
=
1
a
=
a
s
p
+
t
r
=
a
s
p
a
t
r
=
(
a
p
)
s
(
a
r
)
t
=
(
a
p
)
s
1
=
(
a
p
)
s
a
p
∈
H
a
=
(
a
p
)
s
∈
H
a
r
1
p
G
G
p
p
G
G
p
p
p
|
G
|
=
p
n
g
∈
G
p
n
p
|
G
|
p
q
G
q
p
G
n
=
p
1
α
1
⋯
p
k
α
k
p
1
,
…
,
p
k
α
1
,
α
2
,
…
,
α
k
G
G
1
,
G
2
,
…
,
G
k
G
i
G
p
i
r
r
G
G
i
G
i
=
1
,
…
,
k
p
i
0
=
1
1
∈
G
i
g
∈
G
i
p
i
r
g
−
1
p
i
r
h
∈
G
i
p
i
s
(
g
h
)
p
i
t
=
g
p
i
t
h
p
i
t
=
1
⋅
1
=
1
t
r
s
G
=
G
1
G
2
⋯
G
k
G
i
∩
G
j
=
{
1
}
i
j
g
1
∈
G
1
G
2
,
G
3
,
…
,
G
k
g
1
=
g
2
g
3
⋯
g
k
g
i
∈
G
i
g
i
p
α
i
g
i
p
α
i
=
1
i
=
2
,
3
,
…
,
k
g
1
p
2
α
2
⋯
p
k
α
k
=
1
g
1
p
1
gcd
(
p
1
,
p
2
α
2
⋯
p
k
α
k
)
=
1
g
1
=
1
G
1
G
2
,
G
3
,
…
,
G
k
G
i
∩
G
j
=
{
1
}
i
j
g
∈
G
g
1
⋯
g
k
g
i
∈
G
i
g
G
|
g
|
=
p
1
β
1
p
2
β
2
⋯
p
k
β
k
β
1
,
…
,
β
k
a
i
=
|
g
|
/
p
i
β
i
a
i
b
1
,
…
,
b
k
a
1
b
1
+
⋯
+
a
k
b
k
=
1
g
=
g
a
1
b
1
+
⋯
+
a
k
b
k
=
g
a
1
b
1
⋯
g
a
k
b
k
g
(
a
i
b
i
)
p
i
β
i
=
g
b
i
|
g
|
=
e
g
a
i
b
i
G
i
g
i
=
g
a
i
b
i
g
=
g
1
⋯
g
k
∈
G
1
G
2
⋯
G
k
G
=
G
1
G
2
⋯
G
k
p
i
G
i
G
p
g
∈
G
G
⟨
g
⟩
×
H
H
G
G
p
n
n
n
=
1
G
p
g
k
1
≤
k
<
n
g
G
|
g
|
=
p
m
a
p
m
=
e
a
∈
G
h
G
h
∉
⟨
g
⟩
h
h
G
=
⟨
g
⟩
H
=
⟨
h
⟩
⟨
g
⟩
∩
H
=
{
e
}
|
H
|
=
p
|
h
p
|
=
|
h
|
/
p
h
p
h
⟨
g
⟩
h
h
p
=
g
r
r
(
g
r
)
p
m
−
1
=
(
h
p
)
p
m
−
1
=
h
p
m
=
e
g
r
p
m
−
1
g
r
⟨
g
⟩
p
r
r
=
p
s
h
p
=
g
r
=
g
p
s
a
g
−
s
h
a
⟨
g
⟩
h
⟨
g
⟩
a
p
=
g
−
s
p
h
p
=
g
−
r
h
p
=
h
−
p
h
p
=
e
a
p
a
∉
⟨
g
⟩
h
⟨
g
⟩
|
H
|
=
p
g
H
G
/
H
g
G
|
g
H
|
<
|
g
|
=
p
m
H
=
(
g
H
)
p
m
−
1
=
g
p
m
−
1
H
;
g
p
m
−
1
⟨
g
⟩
∩
H
=
{
e
}
g
p
m
g
H
G
/
H
G
/
H
≅
⟨
g
H
⟩
×
K
/
H
K
G
H
⟨
g
⟩
∩
K
=
{
e
}
b
∈
⟨
g
⟩
∩
K
b
H
∈
⟨
g
H
⟩
∩
K
/
H
=
{
H
}
b
∈
⟨
g
⟩
∩
H
=
{
e
}
G
=
⟨
g
⟩
K
G
≅
⟨
g
⟩
×
K
G
g
G
⟨
g
⟩
=
G
G
≅
Z
|
g
|
×
H
H
G
|
H
|
<
|
G
|
G
Z
p
1
α
1
×
Z
p
2
α
2
×
⋯
×
Z
p
n
α
n
×
Z
×
⋯
×
Z
p
i
G
G
=
H
n
⊃
H
n
−
1
⊃
⋯
⊃
H
1
⊃
H
0
=
{
e
}
H
i
H
i
+
1
H
i
G
Z
⊃
9
Z
⊃
45
Z
⊃
180
Z
⊃
{
0
}
,
Z
24
⊃
⟨
2
⟩
⊃
⟨
6
⟩
⊃
⟨
12
⟩
⊃
{
0
}
D
4
D
4
⊃
{
(
1
)
,
(
12
)
(
34
)
,
(
13
)
(
24
)
,
(
14
)
(
23
)
}
⊃
{
(
1
)
,
(
12
)
(
34
)
}
⊃
{
(
1
)
}
{
(
1
)
,
(
12
)
(
34
)
}
D
4
{
K
j
}
{
H
i
}
{
H
i
}
⊂
{
K
j
}
H
i
K
j
Z
⊃
3
Z
⊃
9
Z
⊃
45
Z
⊃
90
Z
⊃
180
Z
⊃
{
0
}
Z
⊃
9
Z
⊃
45
Z
⊃
180
Z
⊃
{
0
}
{
H
i
}
G
H
i
+
1
/
H
i
{
H
i
}
{
K
j
}
G
{
H
i
+
1
/
H
i
}
{
K
j
+
1
/
K
j
}
Z
60
⊃
⟨
3
⟩
⊃
⟨
15
⟩
⊃
{
0
}
Z
60
⊃
⟨
4
⟩
⊃
⟨
20
⟩
⊃
{
0
}
Z
60
Z
60
/
⟨
3
⟩
≅
⟨
20
⟩
/
{
0
}
≅
Z
3
⟨
3
⟩
/
⟨
15
⟩
≅
⟨
4
⟩
/
⟨
20
⟩
≅
Z
5
⟨
15
⟩
/
{
0
}
≅
Z
60
/
⟨
4
⟩
≅
Z
4
{
H
i
}
G
{
H
i
}
G
Z
60
Z
60
⊃
⟨
3
⟩
⊃
⟨
15
⟩
⊃
⟨
30
⟩
⊃
{
0
}
Z
60
/
⟨
3
⟩
≅
Z
3
⟨
3
⟩
/
⟨
15
⟩
≅
Z
5
⟨
15
⟩
/
⟨
30
⟩
≅
Z
2
⟨
30
⟩
/
{
0
}
≅
Z
2
Z
60
Z
60
⊃
⟨
2
⟩
⊃
⟨
4
⟩
⊃
⟨
20
⟩
⊃
{
0
}
n
≥
5
S
n
⊃
A
n
⊃
{
(
1
)
}
S
n
S
n
/
A
n
≅
Z
2
A
n
{
0
}
=
H
0
⊂
H
1
⊂
⋯
⊂
H
n
−
1
⊂
H
n
=
Z
H
1
k
Z
k
∈
N
H
1
/
H
0
≅
k
Z
Z
60
Z
60
Z
2
Z
2
Z
3
Z
5
G
G
k
1
≤
k
<
n
G
=
H
n
⊃
H
n
−
1
⊃
⋯
⊃
H
1
⊃
H
0
=
{
e
}
G
=
K
m
⊃
K
m
−
1
⊃
⋯
⊃
K
1
⊃
K
0
=
{
e
}
G
G
H
i
∩
K
m
−
1
H
i
+
1
∩
K
m
−
1
K
j
∩
H
n
−
1
K
j
+
1
∩
H
n
−
1
G
=
H
n
⊃
H
n
−
1
⊃
H
n
−
1
∩
K
m
−
1
⊃
⋯
⊃
H
0
∩
K
m
−
1
=
{
e
}
G
=
K
m
⊃
K
m
−
1
⊃
K
m
−
1
∩
H
n
−
1
⊃
⋯
⊃
K
0
∩
H
n
−
1
=
{
e
}
H
i
∩
K
m
−
1
H
i
+
1
∩
K
m
−
1
(
H
i
+
1
∩
K
m
−
1
)
/
(
H
i
∩
K
m
−
1
)
=
(
H
i
+
1
∩
K
m
−
1
)
/
(
H
i
∩
(
H
i
+
1
∩
K
m
−
1
)
)
≅
H
i
(
H
i
+
1
∩
K
m
−
1
)
/
H
i
H
i
H
i
(
H
i
+
1
∩
K
m
−
1
)
{
H
i
}
H
i
+
1
/
H
i
H
i
(
H
i
+
1
∩
K
m
−
1
)
/
H
i
H
i
+
1
/
H
i
H
i
/
H
i
H
i
(
H
i
+
1
∩
K
m
−
1
)
H
i
H
i
+
1
H
n
−
1
⊃
H
n
−
1
∩
K
m
−
1
⊃
⋯
⊃
H
0
∩
K
m
−
1
=
{
e
}
H
n
−
1
H
n
−
1
⊃
⋯
⊃
H
1
⊃
H
0
=
{
e
}
G
=
H
n
⊃
H
n
−
1
⊃
⋯
⊃
H
1
⊃
H
0
=
{
e
}
G
=
H
n
⊃
H
n
−
1
⊃
H
n
−
1
∩
K
m
−
1
⊃
⋯
⊃
H
0
∩
K
m
−
1
=
{
e
}
H
n
−
1
=
K
m
−
1
{
H
i
}
{
K
j
}
H
n
−
1
K
m
−
1
G
H
n
−
1
H
n
−
1
K
m
−
1
=
G
K
m
−
1
/
(
K
m
−
1
∩
H
n
−
1
)
≅
(
H
n
−
1
K
m
−
1
)
/
H
n
−
1
=
G
/
H
n
−
1
G
=
H
n
⊃
H
n
−
1
⊃
H
n
−
1
∩
K
m
−
1
⊃
⋯
⊃
H
0
∩
K
m
−
1
=
{
e
}
G
=
K
m
⊃
K
m
−
1
⊃
K
m
−
1
∩
H
n
−
1
⊃
⋯
⊃
K
0
∩
H
n
−
1
=
{
e
}
G
{
H
i
}
H
i
+
1
/
H
i
S
4
S
4
⊃
A
4
⊃
{
(
1
)
,
(
12
)
(
34
)
,
(
13
)
(
24
)
,
(
14
)
(
23
)
}
⊃
{
(
1
)
}
n
≥
5
S
n
⊃
A
n
⊃
{
(
1
)
}
S
n
S
n
n
≥
5
16
200
=
2
3
5
2
729
=
3
6
Z
8
×
Z
3
×
Z
3
72
8
G
40
200
720
Z
12
Z
48
Q
8
D
4
S
3
×
Z
4
S
4
S
n
n
≥
5
Q
{
0
}
⊂
⟨
6
⟩
⊂
⟨
3
⟩
⊂
Z
12
{
(
1
)
}
×
{
0
}
⊂
{
(
1
)
,
(
123
)
,
(
132
)
}
×
{
0
}
⊂
S
3
×
{
0
}
⊂
S
3
×
⟨
2
⟩
⊂
S
3
×
Z
4
G
=
Z
2
×
Z
2
×
⋯
G
m
n
m
G
n
G
G
G
H
K
G
×
H
≅
G
×
K
H
≅
K
G
H
G
×
H
G
N
G
N
N
G
N
G
/
N
G
N
G
N
G
/
N
G
N
G
/
N
N
=
N
n
⊃
N
n
−
1
⊃
⋯
⊃
N
1
⊃
N
0
=
{
e
}
G
/
N
=
G
n
/
N
⊃
G
n
−
1
/
N
⊃
⋯
G
1
/
N
⊃
G
0
/
N
=
{
N
}
G
G
G
=
P
n
⊃
P
n
−
1
⊃
⋯
⊃
P
1
⊃
P
0
=
{
e
}
P
i
P
i
+
1
P
i
+
1
/
P
i
G
G
G
N
G
G
/
N
D
n
n
D
n
2
G
N
G
N
G
/
N
G
p
H
K
H
K
K
H
G
n
≥
2
G
G
′
G
G
a
−
1
b
−
1
a
b
a
,
b
∈
G
G
G
(
0
)
=
G
G
(
1
)
=
G
′
G
(
i
+
1
)
=
(
G
(
i
)
)
′
G
(
i
+
1
)
(
G
(
i
)
)
′
G
(
0
)
=
G
⊃
G
(
1
)
⊃
G
(
2
)
⊃
⋯
G
G
G
(
n
)
=
{
e
}
n
G
n
≥
2
G
G
/
G
′
H
K
G
H
∗
K
∗
H
K
H
∗
(
H
∩
K
∗
)
H
∗
(
H
∩
K
)
K
∗
(
H
∗
∩
K
)
K
∗
(
H
∩
K
)
H
∗
(
H
∩
K
)
/
H
∗
(
H
∩
K
∗
)
≅
K
∗
(
H
∩
K
)
/
K
∗
(
H
∗
∩
K
)
≅
(
H
∩
K
)
/
(
H
∗
∩
K
)
(
H
∩
K
∗
)
G
n
n
16
2
p
p
2
p
2
p
p
n
=
6
,
10
,
14
n
=
9
p
2
p
Z
9
Z
3
×
Z
3
n
=
15
Z
3
×
Z
5
≅
Z
15
n
=
8
n
=
12
n
=
16
14
n
=
8
3
n
=
12
2
3
4
4
12
8
2
k
k
>
2
16
16
Z
3
⋊
Z
4
G
X
g
∈
G
x
∈
X
g
x
X
X
G
G
X
G
×
X
→
X
(
g
,
x
)
↦
g
x
e
x
=
x
x
∈
X
(
g
1
g
2
)
x
=
g
1
(
g
2
x
)
x
∈
X
g
1
,
g
2
∈
G
X
G
G
X
G
G
X
(
g
,
x
)
↦
x
X
G
G
=
G
L
2
(
R
)
X
=
R
2
G
X
v
∈
R
2
I
I
v
=
v
A
B
2
×
2
(
A
B
)
v
=
A
(
B
v
)
G
=
D
4
X
=
{
1
,
2
,
3
,
4
}
D
4
{
(
1
)
,
(
13
)
,
(
24
)
,
(
1432
)
,
(
1234
)
,
(
12
)
(
34
)
,
(
14
)
(
23
)
,
(
13
)
(
24
)
}
D
4
X
(
13
)
(
24
)
1
3
2
4
X
G
S
X
X
X
G
(
σ
,
x
)
↦
σ
(
x
)
σ
∈
G
x
∈
X
X
=
G
G
(
g
,
x
)
↦
λ
g
(
x
)
=
g
x
λ
g
e
⋅
x
=
λ
e
x
=
e
x
=
x
(
g
h
)
⋅
x
=
λ
g
h
x
=
λ
g
λ
h
x
=
λ
g
(
h
x
)
=
g
⋅
(
h
⋅
x
)
H
G
G
H
H
G
X
=
G
H
G
G
H
H
G
H
×
G
→
G
(
h
,
g
)
↦
h
g
h
−
1
h
∈
H
g
∈
G
(
h
1
h
2
,
g
)
=
h
1
h
2
g
(
h
1
h
2
)
−
1
=
h
1
(
h
2
g
h
2
−
1
)
h
1
−
1
=
(
h
1
,
(
h
2
,
g
)
)
H
G
L
H
H
L
H
G
(
g
,
x
H
)
↦
g
x
H
(
g
g
′
)
x
H
=
g
(
g
′
x
H
)
G
X
x
,
y
∈
X
x
G
G
y
g
∈
G
g
x
=
y
x
∼
G
y
x
∼
y
G
X
G
G
X
∼
e
x
=
x
x
∼
y
x
,
y
∈
X
g
g
x
=
y
g
−
1
y
=
x
y
∼
x
x
∼
y
y
∼
z
g
h
g
x
=
y
h
y
=
z
z
=
h
y
=
(
h
g
)
x
x
z
X
G
X
G
X
G
x
X
O
x
x
G
G
=
{
(
1
)
,
(
1
2
3
)
,
(
1
3
2
)
,
(
4
5
)
,
(
1
2
3
)
(
4
5
)
,
(
1
3
2
)
(
4
5
)
}
X
=
{
1
,
2
,
3
,
4
,
5
}
X
G
O
1
=
O
2
=
O
3
=
{
1
,
2
,
3
}
O
4
=
O
5
=
{
4
,
5
}
G
X
g
G
g
X
X
g
x
∈
X
g
x
=
x
g
g
x
∈
X
G
x
x
G
x
x
X
g
⊂
X
G
x
⊂
G
X
=
{
1
,
2
,
3
,
4
,
5
,
6
}
G
{
(
1
)
,
(
1
2
)
(
3
4
5
6
)
,
(
3
5
)
(
4
6
)
,
(
1
2
)
(
3
6
5
4
)
}
X
G
X
(
1
)
=
X
,
X
(
3
5
)
(
4
6
)
=
{
1
,
2
}
,
X
(
1
2
)
(
3
4
5
6
)
=
X
(
1
2
)
(
3
6
5
4
)
=
∅
G
1
=
G
2
=
{
(
1
)
,
(
3
5
)
(
4
6
)
}
,
G
3
=
G
4
=
G
5
=
G
6
=
{
(
1
)
}
G
x
G
x
∈
X
G
X
x
∈
X
x
G
x
G
e
∈
G
x
X
g
,
h
∈
G
x
g
x
=
x
h
x
=
x
(
g
h
)
x
=
g
(
h
x
)
=
g
x
=
x
G
x
G
x
g
∈
G
x
x
=
e
x
=
(
g
−
1
g
)
x
=
(
g
−
1
)
g
x
=
g
−
1
x
g
−
1
G
x
g
∈
G
|
X
g
|
x
∈
X
|
O
x
|
x
∈
X
G
x
G
G
X
G
x
∈
X
|
O
x
|
=
[
G
:
G
x
]
|
G
|
/
|
G
x
|
G
x
G
ϕ
O
x
X
L
G
x
G
x
G
y
∈
O
x
g
G
g
x
=
y
ϕ
ϕ
(
y
)
=
g
G
x
ϕ
ϕ
(
y
1
)
=
ϕ
(
y
2
)
ϕ
(
y
1
)
=
g
1
G
x
=
g
2
G
x
=
ϕ
(
y
2
)
g
1
x
=
y
1
g
2
x
=
y
2
g
1
G
x
=
g
2
G
x
g
∈
G
x
g
2
=
g
1
g
y
2
=
g
2
x
=
g
1
g
x
=
g
1
x
=
y
1
;
ϕ
ϕ
g
G
x
g
x
=
y
ϕ
(
y
)
=
g
G
x
X
G
X
G
X
X
G
=
{
x
∈
X
:
g
x
=
x
for all
g
∈
G
}
X
|
X
|
=
|
X
G
|
+
∑
i
=
k
n
|
O
x
i
|
x
k
,
…
,
x
n
X
G
(
g
,
x
)
↦
g
x
g
−
1
G
Z
(
G
)
=
{
x
:
x
g
=
g
x
for all
g
∈
G
}
G
x
1
,
…
,
x
k
G
|
O
x
1
|
=
n
1
,
…
,
|
O
x
k
|
=
n
k
|
G
|
=
|
Z
(
G
)
|
+
n
1
+
⋯
+
n
k
x
i
C
(
x
i
)
=
{
g
∈
G
:
g
x
i
=
x
i
g
}
x
i
|
G
|
=
|
Z
(
G
)
|
+
[
G
:
C
(
x
1
)
]
+
⋯
+
[
G
:
C
(
x
k
)
]
G
S
3
{
(
1
)
}
,
{
(
123
)
,
(
132
)
}
,
{
(
12
)
,
(
13
)
,
(
23
)
}
6
=
1
+
2
+
3
D
4
{
(
1
)
,
(
13
)
(
24
)
}
{
(
13
)
,
(
24
)
}
,
{
(
1432
)
,
(
1234
)
}
,
{
(
12
)
(
34
)
,
(
14
)
(
23
)
}
D
4
8
=
2
+
2
+
2
+
2
S
n
σ
=
(
a
1
,
…
,
a
k
)
τ
∈
S
n
τ
σ
τ
−
1
=
(
τ
(
a
1
)
,
…
,
τ
(
a
k
)
)
σ
=
σ
1
σ
2
⋯
σ
r
σ
i
r
i
σ
τ
∈
S
n
S
n
n
S
3
3
3
=
1
+
1
+
1
3
=
1
+
2
3
=
3
;
n
n
G
p
n
p
G
|
G
|
=
|
Z
(
G
)
|
+
n
1
+
⋯
+
n
k
n
i
>
1
n
i
∣
|
G
|
p
n
i
p
∣
|
G
|
p
|
Z
(
G
)
|
G
|
Z
(
G
)
|
≥
1
|
Z
(
G
)
|
≥
p
g
∈
Z
(
G
)
g
1
G
p
2
p
G
|
Z
(
G
)
|
=
p
p
2
|
Z
(
G
)
|
=
p
2
|
Z
(
G
)
|
=
p
Z
(
G
)
G
/
Z
(
G
)
p
a
Z
(
G
)
G
/
Z
(
G
)
g
Z
(
G
)
a
m
Z
(
G
)
m
g
=
a
m
x
x
G
h
Z
(
G
)
∈
G
/
Z
(
G
)
y
Z
(
G
)
h
=
a
n
y
n
x
y
G
G
g
h
=
a
m
x
a
n
y
=
a
m
+
n
x
y
=
a
n
y
a
m
x
=
h
g
G
2
4
=
16
90
∘
X
G
x
∼
y
G
x
G
y
|
G
x
|
=
|
G
y
|
G
X
(
g
,
x
)
↦
g
⋅
x
x
∼
y
g
∈
G
g
⋅
x
=
y
a
∈
G
x
g
a
g
−
1
⋅
y
=
g
a
⋅
g
−
1
y
=
g
a
⋅
x
=
g
⋅
x
=
y
ϕ
:
G
x
→
G
y
ϕ
(
a
)
=
g
a
g
−
1
ϕ
ϕ
(
a
b
)
=
g
a
b
g
−
1
=
g
a
g
−
1
g
b
g
−
1
=
ϕ
(
a
)
ϕ
(
b
)
ϕ
(
a
)
=
ϕ
(
b
)
g
a
g
−
1
=
g
b
g
−
1
a
=
b
ϕ
b
G
y
g
−
1
b
g
G
x
g
−
1
b
g
⋅
x
=
g
−
1
b
⋅
g
x
=
g
−
1
b
⋅
y
=
g
−
1
⋅
y
=
x
;
ϕ
(
g
−
1
b
g
)
=
b
G
X
k
X
k
=
1
|
G
|
∑
g
∈
G
|
X
g
|
x
g
∈
G
g
x
g
x
=
x
g
x
∑
g
∈
G
|
X
g
|
∑
x
∈
X
|
G
x
|
;
∑
g
∈
G
|
X
g
|
=
∑
x
∈
X
|
G
x
|
∑
y
∈
O
x
|
G
y
|
=
|
O
x
|
⋅
|
G
x
|
|
G
|
k
∑
g
∈
G
|
X
g
|
=
∑
x
∈
X
|
G
x
|
=
k
⋅
|
G
|
X
=
{
1
,
2
,
3
,
4
,
5
}
G
G
=
{
(
1
)
,
(
1
3
)
,
(
1
3
)
(
2
5
)
,
(
2
5
)
}
X
{
1
,
3
}
{
2
,
5
}
{
4
}
X
(
1
)
=
X
X
(
1
3
)
=
{
2
,
4
,
5
}
X
(
1
3
)
(
2
5
)
=
{
4
}
X
(
2
5
)
=
{
1
,
3
,
4
}
k
=
1
|
G
|
∑
g
∈
G
|
X
g
|
=
1
4
(
5
+
3
+
1
+
3
)
=
3
D
4
(
1
)
(
13
)
(
24
)
(
1432
)
(
1234
)
(
12
)
(
34
)
(
14
)
(
23
)
(
13
)
(
24
)
G
{
1
,
2
,
3
,
4
}
X
Y
=
{
B
,
W
}
B
W
f
:
X
→
Y
σ
∈
D
4
σ
~
σ
~
(
f
)
=
f
∘
σ
f
:
X
→
Y
f
f
(
1
)
=
B
f
(
2
)
=
W
f
(
3
)
=
W
f
(
4
)
=
W
σ
=
(
1
2
)
(
3
4
)
σ
~
(
f
)
=
f
∘
σ
2
B
W
σ
~
G
~
X
~
X
~
X
Y
G
~
X
~
(
1
)
=
X
~
|
X
~
|
=
2
4
=
16
X
~
(
1
2
3
4
)
f
∈
X
~
f
(
1
23
4
)
f
(
1
)
=
f
(
2
)
=
f
(
3
)
=
f
(
4
)
f
f
(
x
)
=
B
f
(
x
)
=
W
x
|
X
~
(
1
2
3
4
)
|
=
2
|
X
~
(
1
4
3
2
)
|
=
2
X
~
(
1
3
)
(
2
4
)
f
(
1
)
=
f
(
3
)
f
(
2
)
=
f
(
4
)
|
X
~
(
13
)
(
24
)
|
=
2
2
=
4
|
X
~
(
1
2
)
(
3
4
)
|
=
4
|
X
~
(
1
4
)
(
2
3
)
|
=
4
X
~
(
1
3
)
f
(
1
)
=
f
(
3
)
|
X
~
(
1
3
)
|
=
2
3
=
8
|
X
~
(
2
4
)
|
=
8
1
8
(
2
4
+
2
1
+
2
2
+
2
1
+
2
2
+
2
2
+
2
3
+
2
3
)
=
6
G
X
X
~
X
Y
G
~
X
~
σ
~
∈
G
~
σ
~
(
f
)
=
f
∘
σ
σ
∈
G
f
∈
X
~
n
σ
|
X
~
σ
|
=
|
Y
|
n
σ
∈
G
f
∈
X
~
f
∘
σ
X
~
g
X
Y
σ
~
(
f
)
=
σ
~
(
g
)
x
∈
X
f
(
σ
(
x
)
)
=
σ
~
(
f
)
(
x
)
=
σ
~
(
g
)
(
x
)
=
g
(
σ
(
x
)
)
σ
X
x
′
X
x
X
σ
f
g
X
f
=
g
σ
~
σ
↦
σ
~
σ
X
σ
=
σ
1
σ
2
⋯
σ
n
f
X
~
σ
σ
n
|
Y
|
|
X
~
σ
|
=
|
Y
|
n
X
=
{
1
,
2
,
…
,
7
}
Y
=
{
A
,
B
,
C
}
g
X
(
1
3
)
(
2
4
5
)
=
(
1
3
)
(
2
4
5
)
(
6
)
(
7
)
n
=
4
f
∈
X
~
g
g
|
Y
|
=
3
|
X
~
g
|
=
3
4
=
81
1
8
(
4
4
+
4
1
+
4
2
+
4
1
+
4
2
+
4
2
+
4
3
+
4
3
)
=
55
n
n
n
Z
2
n
Z
2
n
2
n
n
2
2
n
n
a
b
f
g
g
f
g
(
a
,
b
,
c
)
=
f
(
b
,
c
,
a
)
g
∼
f
(
a
c
b
)
(
a
b
)
f
2
∼
f
4
f
3
∼
f
5
f
10
∼
f
12
f
11
∼
f
13
f
0
f
1
f
2
f
3
f
4
f
5
f
6
f
7
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
1
1
1
1
0
0
0
1
1
0
0
1
1
1
1
0
1
0
1
0
1
0
1
f
8
f
9
f
10
f
11
f
12
f
13
f
14
f
15
0
0
1
1
1
1
1
1
1
1
0
1
0
0
0
0
1
1
1
1
1
0
0
0
1
1
0
0
1
1
1
1
0
1
0
1
0
1
0
1
2
2
3
=
256
2
2
4
=
65,536
a
b
c
f
g
f
g
{
a
,
b
,
c
}
a
b
c
2
3
(
a
,
b
,
c
)
(
a
c
b
)
(
0
,
0
,
0
)
↦
(
0
,
0
,
0
)
(
0
,
0
,
1
)
↦
(
0
,
1
,
0
)
(
0
,
1
,
0
)
↦
(
1
,
0
,
0
)
⋮
(
1
,
1
,
0
)
↦
(
1
,
0
,
1
)
(
1
,
1
,
1
)
↦
(
1
,
1
,
1
)
X
n
|
X
|
=
2
n
(
0
,
…
,
0
,
1
)
↦
0
(
0
,
…
,
1
,
0
)
↦
1
(
0
,
…
,
1
,
1
)
↦
2
⋮
(
1
,
…
,
1
,
1
)
↦
2
n
−
1
(
a
)
,
(
a
c
)
,
(
b
d
)
,
(
a
d
c
b
)
,
(
a
b
c
d
)
,
(
a
b
)
(
c
d
)
,
(
a
d
)
(
b
c
)
,
(
a
c
)
(
b
d
)
1
8
(
2
16
+
2
⋅
2
12
+
2
⋅
2
6
+
3
⋅
2
10
)
=
9616
(
a
)
(
0
)
(
a
c
)
(
2
,
8
)
(
3
,
9
)
(
6
,
12
)
(
7
,
13
)
(
b
d
)
(
1
,
4
)
(
3
,
6
)
(
9
,
12
)
(
11
,
14
)
(
a
d
c
b
)
(
1
,
2
,
4
,
8
)
(
3
,
6.12
,
9
)
(
5
,
10
)
(
7
,
14
,
13
,
11
)
(
a
b
c
d
)
(
1
,
8
,
4
,
2
)
(
3
,
9
,
12
,
6
)
(
5
,
10
)
(
7
,
11
,
13
,
14
)
(
a
b
)
(
c
d
)
(
1
,
2
)
(
4
,
8
)
(
5
,
10
)
(
6
,
9
)
(
7
,
11
)
(
13
,
14
)
(
a
d
)
(
b
c
)
(
1
,
8
)
(
2
,
4
)
(
3
,
12
)
(
5
,
10
)
(
7
,
14
)
(
11
,
13
)
(
a
c
)
(
b
d
)
(
1
,
4
)
(
2
,
8
)
(
3
,
12
)
(
6
,
9
)
(
7
,
13
)
(
11
,
14
)
G
G
=
H
n
⊃
H
n
−
1
⊃
⋯
⊃
H
1
⊃
H
0
=
{
e
}
H
i
H
i
+
1
H
i
+
1
/
H
i
49
G
X
G
G
0
R
2
∖
{
0
}
X
=
{
1
,
2
,
3
,
4
}
X
g
G
x
X
=
{
1
,
2
,
3
}
G
=
S
3
=
{
(
1
)
,
(
12
)
,
(
13
)
,
(
23
)
,
(
123
)
,
(
132
)
}
X
=
{
1
,
2
,
3
,
4
,
5
,
6
}
G
=
{
(
1
)
,
(
12
)
,
(
345
)
,
(
354
)
,
(
12
)
(
345
)
,
(
12
)
(
354
)
}
X
(
1
)
=
{
1
,
2
,
3
}
X
(
12
)
=
{
3
}
X
(
13
)
=
{
2
}
X
(
23
)
=
{
1
}
X
(
123
)
=
X
(
132
)
=
∅
G
1
=
{
(
1
)
,
(
23
)
}
G
2
=
{
(
1
)
,
(
13
)
}
G
3
=
{
(
1
)
,
(
12
)
}
G
X
G
x
∈
X
|
G
|
=
|
O
x
|
⋅
|
G
x
|
O
1
=
O
2
=
O
3
=
{
1
,
2
,
3
}
G
θ
∈
G
R
2
θ
P
R
2
G
P
G
P
G
=
A
4
G
(
g
,
h
)
↦
g
h
g
−
1
G
G
S
4
D
5
Z
9
Q
8
S
4
O
(
1
)
=
{
(
1
)
}
,
O
(
12
)
=
{
(
12
)
,
(
13
)
,
(
14
)
,
(
23
)
,
(
24
)
,
(
34
)
}
,
O
(
12
)
(
34
)
=
{
(
12
)
(
34
)
,
(
13
)
(
24
)
,
(
14
)
(
23
)
}
,
O
(
123
)
=
{
(
123
)
,
(
132
)
,
(
124
)
,
(
142
)
,
(
134
)
,
(
143
)
,
(
234
)
,
(
243
)
}
,
O
(
1234
)
=
{
(
1234
)
,
(
1243
)
,
(
1324
)
,
(
1342
)
,
(
1423
)
,
(
1432
)
}
1
+
3
+
6
+
6
+
8
=
24
S
5
A
5
(
3
4
+
3
1
+
3
2
+
3
1
+
3
2
+
3
2
+
3
3
+
3
3
)
/
8
=
21
1
,
…
,
6
S
4
(
a
b
c
d
)
(
a
b
)
(
c
d
)
(
a
b
)
(
c
d
)
(
e
f
)
(
a
b
c
)
(
d
e
f
)
12
(
1
⋅
2
6
+
3
⋅
2
4
+
4
⋅
2
3
+
2
⋅
2
2
+
2
⋅
2
1
)
/
12
=
13
C
H
3
x
1
x
2
x
3
S
3
x
1
x
2
x
3
x
4
S
4
(
1
⋅
2
8
+
3
⋅
2
6
+
2
⋅
2
4
)
/
6
=
80
x
1
x
2
x
3
x
4
S
4
(
x
1
x
2
x
3
x
4
)
12
G
X
G
X
G
X
G
X
p
p
n
S
n
a
∈
G
g
∈
G
g
C
(
a
)
g
−
1
=
C
(
g
a
g
−
1
)
x
∈
g
C
(
a
)
g
−
1
g
−
1
x
g
∈
C
(
a
)
|
G
|
=
p
n
p
|
Z
(
G
)
|
<
p
n
−
1
G
p
n
p
X
G
X
G
=
{
x
∈
X
:
g
x
=
x
for all
g
∈
G
}
X
|
X
|
≡
|
X
G
|
(
mod
p
)
G
p
n
p
n
≥
2
G
p
n
≥
3
G
p
2
S
n
n
g
x
g
−
1
g
−
1
x
g
4
4
16
0
1
24
48
4
000
,
010
,
110
,
100
001
,
011
,
111
,
101
000
6
48
8
000
100
010
001
000
3
!
=
6
11
2
2
10
11
3
S
5
A
5
7
8
D
7
D
8
Q
4
16
G
m
n
m
G
n
A
4
12
6
G
G
(
g
,
x
)
↦
g
x
g
−
1
x
1
,
…
,
x
k
G
|
G
|
=
|
Z
(
G
)
|
+
[
G
:
C
(
x
1
)
]
+
⋯
+
[
G
:
C
(
x
k
)
]
Z
(
G
)
=
{
g
∈
G
:
g
x
=
x
g
for all
x
∈
G
}
G
C
(
x
i
)
=
{
g
∈
G
:
g
x
i
=
x
i
g
}
x
i
p
p
G
p
p
G
p
p
G
p
p
p
G
p
p
G
G
p
G
|
G
|
=
p
G
k
p
≤
k
<
n
p
k
p
|
G
|
=
n
p
∣
n
G
|
G
|
=
|
Z
(
G
)
|
+
[
G
:
C
(
x
1
)
]
+
⋯
+
[
G
:
C
(
x
k
)
]
C
(
x
i
)
p
i
i
=
1
,
…
,
k
C
(
x
i
)
G
p
|
C
(
x
i
)
|
C
(
x
i
)
p
G
p
p
p
[
G
:
C
(
x
i
)
]
p
G
Z
(
G
)
Z
(
G
)
p
G
p
G
G
p
|
G
|
=
p
n
A
5
|
A
5
|
=
60
=
2
2
⋅
3
⋅
5
A
5
2
3
5
A
5
G
p
p
r
|
G
|
G
p
r
G
|
G
|
=
p
G
n
n
>
p
n
p
n
|
G
|
=
|
Z
(
G
)
|
+
[
G
:
C
(
x
1
)
]
+
⋯
+
[
G
:
C
(
x
k
)
]
p
[
G
:
C
(
x
i
)
]
i
p
r
∣
|
C
(
x
i
)
|
p
r
|
G
|
=
|
C
(
x
i
)
|
⋅
[
G
:
C
(
x
i
)
]
C
(
x
i
)
p
[
G
:
C
(
x
i
)
]
i
p
|
G
|
p
|
Z
(
G
)
|
Z
(
G
)
p
g
N
g
N
Z
(
G
)
Z
(
G
)
N
G
Z
(
G
)
G
G
/
N
|
G
|
/
p
G
/
N
H
p
r
−
1
H
ϕ
:
G
→
G
/
N
p
r
G
p
p
p
P
G
p
G
G
S
G
H
S
H
H
S
H
×
S
→
S
h
⋅
K
↦
h
K
h
−
1
K
S
H
N
(
H
)
=
{
g
∈
G
:
g
H
g
−
1
=
H
}
G
H
G
H
N
(
H
)
N
(
H
)
G
H
P
p
G
x
p
x
−
1
P
x
=
P
x
∈
P
x
∈
N
(
P
)
⟨
x
P
⟩
⊂
N
(
P
)
/
P
p
H
N
(
P
)
P
H
/
P
=
⟨
x
P
⟩
|
H
|
=
|
P
|
⋅
|
⟨
x
P
⟩
|
H
p
P
p
H
P
p
|
G
|
H
=
P
H
/
P
x
P
=
P
x
∈
P
H
K
G
H
K
[
H
:
N
(
K
)
∩
H
]
K
N
(
K
)
∩
H
h
−
1
K
h
↦
(
N
(
K
)
∩
H
)
h
h
1
,
h
2
∈
H
(
N
(
K
)
∩
H
)
h
1
=
(
N
(
K
)
∩
H
)
h
2
h
2
h
1
−
1
∈
N
(
K
)
K
=
h
2
h
1
−
1
K
h
1
h
2
−
1
h
1
−
1
K
h
1
=
h
2
−
1
K
h
2
H
K
N
(
K
)
∩
H
H
G
p
|
G
|
p
G
P
1
P
2
p
g
∈
G
g
P
1
g
−
1
=
P
2
P
p
G
|
G
|
=
p
r
m
|
P
|
=
p
r
S
=
{
P
=
P
1
,
P
2
,
…
,
P
k
}
P
G
k
=
[
G
:
N
(
P
)
]
|
G
|
=
p
r
m
=
|
N
(
P
)
|
⋅
[
G
:
N
(
P
)
]
=
|
N
(
P
)
|
⋅
k
p
r
|
N
(
P
)
|
p
k
p
Q
Q
∈
S
Q
P
i
S
P
i
[
Q
:
N
(
P
i
)
∩
Q
]
|
Q
|
=
[
Q
:
N
(
P
i
)
∩
Q
]
|
N
(
P
i
)
∩
Q
|
[
Q
:
N
(
P
i
)
∩
Q
]
|
Q
|
=
p
r
p
p
k
p
P
j
x
−
1
P
j
x
=
P
j
x
∈
Q
P
j
=
Q
G
p
G
p
1
(
mod
p
)
|
G
|
P
p
p
S
=
{
P
=
P
1
,
P
2
,
…
,
P
k
}
P
P
P
p
P
p
|
S
|
{
P
}
|
S
|
p
1
|
S
|
≡
1
(
mod
p
)
G
S
p
P
∈
S
|
S
|
=
|
orbit of
P
|
=
[
G
:
N
(
P
)
]
[
G
:
N
(
P
)
]
|
G
|
p
A
5
2
3
4
5
p
A
5
3
4
5
p
A
5
5
60
1
(
mod
5
)
5
A
5
5
5
A
5
A
5
5
A
5
p
q
p
<
q
G
p
q
q
G
G
q
≢
1
(
mod
p
)
G
G
H
q
H
p
q
1
+
k
q
k
=
0
,
1
,
…
1
+
q
H
H
G
G
p
K
K
q
1
+
k
p
k
=
0
,
1
,
…
q
1
+
k
p
=
q
1
+
k
p
=
1
1
+
k
p
=
1
K
G
G
H
K
H
Z
q
K
Z
p
G
≅
Z
p
×
Z
q
≅
Z
p
q
15
15
=
5
⋅
3
5
≢
1
(
mod
3
)
99
=
3
2
⋅
11
G
99
1
+
3
k
3
9
k
=
0
,
1
,
2
,
…
1
+
3
k
11
3
H
G
1
+
11
k
11
1
+
11
k
9
11
K
G
p
2
p
H
Z
3
×
Z
3
Z
9
K
11
Z
11
99
Z
3
×
Z
3
×
Z
11
Z
9
×
Z
11
5
⋅
7
⋅
47
=
1645
G
′
=
⟨
a
b
a
−
1
b
−
1
:
a
,
b
∈
G
⟩
a
b
a
−
1
b
−
1
G
G
′
G
G
/
G
′
G
′
G
G
5
⋅
7
⋅
47
=
1645
G
H
1
47
G
/
H
1
G
H
|
G
′
|
1
47
|
G
′
|
=
1
|
G
′
|
=
47
G
5
7
H
2
H
3
G
|
H
2
|
=
5
|
H
3
|
=
7
G
′
H
i
i
=
1
,
2
G
′
1
5
7
|
G
′
|
=
1
47
G
G
A
5
G
20
G
5
1
(
mod
5
)
20
1
5
5
G
p
n
n
>
1
p
G
G
G
4
8
9
16
25
27
32
49
64
81
4
9
25
49
56
=
2
3
⋅
7
p
p
7
7
7
8
⋅
6
=
48
7
2
2
2
2
48
7
7
2
8
2
56
2
G
G
G
48
48
H
K
G
|
H
K
|
=
|
H
|
⋅
|
K
|
|
H
∩
K
|
H
K
=
{
h
k
:
h
∈
H
,
k
∈
K
}
|
H
K
|
≤
|
H
|
⋅
|
K
|
H
K
H
K
h
1
k
1
=
h
2
k
2
h
1
,
h
2
∈
H
k
1
,
k
2
∈
K
a
=
(
h
1
)
−
1
h
2
=
k
1
(
k
2
)
−
1
a
∈
H
∩
K
(
h
1
)
−
1
h
2
H
k
2
(
k
1
)
−
1
K
h
2
=
h
1
a
−
1
k
2
=
a
k
1
h
=
h
1
b
−
1
k
=
b
k
1
b
∈
H
∩
K
h
k
=
h
1
k
1
h
∈
H
k
∈
K
h
k
∈
H
K
h
i
k
i
h
i
∈
H
k
i
∈
K
H
∩
K
|
H
∩
K
|
|
H
K
|
=
(
|
H
|
⋅
|
K
|
)
/
|
H
∩
K
|
G
48
G
8
16
G
2
16
2
H
K
|
H
∩
K
|
=
8
|
H
∩
K
|
≤
4
|
H
K
|
≥
16
⋅
16
4
=
64
H
∩
K
H
K
H
K
H
∩
K
H
N
(
H
∩
K
)
N
(
H
∩
K
)
16
|
N
(
H
∩
K
)
|
16
1
48
|
N
(
H
∩
K
)
|
=
48
N
(
H
∩
K
)
=
G
p
p
p
69
G
G
p
G
18
24
54
72
80
|
G
|
=
18
=
2
⋅
3
2
2
2
3
9
3
S
4
3
S
4
P
1
=
{
(
1
)
,
(
123
)
,
(
132
)
}
P
2
=
{
(
1
)
,
(
124
)
,
(
142
)
}
P
3
=
{
(
1
)
,
(
134
)
,
(
143
)
}
P
4
=
{
(
1
)
,
(
234
)
,
(
243
)
}
45
9
H
p
G
H
p
G
N
(
H
)
96
|
G
|
=
96
=
2
5
⋅
3
G
2
2
H
K
|
H
∩
K
|
≥
16
H
K
(
32
⋅
32
)
/
8
=
128
H
∩
K
H
K
2
160
H
G
|
H
|
=
p
k
p
H
p
G
G
p
2
q
2
p
q
q
∤
p
2
−
1
p
∤
q
2
−
1
G
G
p
p
2
q
q
2
33
3
H
G
H
G
G
p
p
G
G
G
p
r
p
G
p
r
−
1
G
p
n
k
k
<
p
G
H
G
g
N
(
H
)
g
−
1
=
N
(
g
H
g
−
1
)
g
∈
G
108
175
255
G
G
|
G
|
=
3
⋅
5
⋅
17
G
p
1
e
1
⋯
p
n
e
n
G
n
p
P
1
,
…
,
P
n
|
P
i
|
=
p
i
e
i
G
P
1
×
⋯
×
P
n
P
p
G
G
P
G
G
|
G
|
H
G
P
p
H
g
∈
G
h
H
g
P
g
−
1
=
h
P
h
−
1
N
P
G
=
H
N
G
p
n
q
p
q
p
>
q
G
H
G
[
G
:
N
(
H
)
]
N
(
H
)
G
H
G
N
(
H
)
g
↦
g
−
1
H
g
2
S
5
D
4
p
p
m
p
∤
(
p
k
m
p
k
)
S
p
k
G
p
|
S
|
G
S
a
T
=
{
a
t
:
t
∈
T
}
a
∈
G
T
∈
S
p
∤
|
O
T
|
T
∈
S
{
T
1
,
…
,
T
u
}
p
∤
u
H
=
{
g
∈
G
:
g
T
1
=
T
1
}
H
G
|
G
|
=
u
|
H
|
p
k
|
H
|
p
k
≤
|
H
|
|
H
|
=
|
O
T
|
≤
p
k
p
k
=
|
H
|
G
G
′
=
⟨
a
b
a
−
1
b
−
1
:
a
,
b
∈
G
⟩
G
G
/
G
′
{
a
b
a
−
1
b
−
1
:
a
,
b
∈
G
}
a
G
′
,
b
G
′
∈
G
/
G
′
(
a
G
′
)
(
b
G
′
)
=
a
b
G
′
=
a
b
(
b
−
1
a
−
1
b
a
)
G
′
=
(
a
b
b
−
1
a
−
1
)
b
a
G
′
=
b
a
G
′
60
G
|
G
|
≤
60
1
16
14
31
1
46
2
2
17
1
32
51
47
1
3
18
33
1
48
52
4
19
34
49
5
20
5
35
1
50
5
6
21
36
14
51
7
22
2
37
1
52
8
23
1
38
53
9
24
39
2
54
15
10
25
2
40
14
55
2
11
26
2
41
1
56
12
5
27
5
42
57
2
13
28
43
1
58
14
29
1
44
4
59
1
15
1
30
4
45
60
13
G
|
G
|
≤
60
G
G
n
n
=
1
,
…
,
60
n
p
p
0
p
p
p
p
D
18
36
=
2
2
⋅
3
2
2
4
3
9
p
=
2
2
p
=
2
1
,
3
9
2
9
18
2
4
p
=
3
1
4
3
3
3
D
18
3
3
6
=
2
⋅
3
D
18
D
18
6
64
64
2
26
H
S
2
5
44
352
000
40
D
20
p
A
5
A
5
D
36
36
p
p
72
G
36
D
18
H
3
K
6
K
H
H
K
48
4
n
n
=
2
p
p
5
A
5
5
A
5
5
S
6
A
5
5
1
p
R
a
+
b
=
b
+
a
a
,
b
∈
R
(
a
+
b
)
+
c
=
a
+
(
b
+
c
)
a
,
b
,
c
∈
R
0
R
a
+
0
=
a
a
∈
R
a
∈
R
−
a
R
a
+
(
−
a
)
=
0
(
a
b
)
c
=
a
(
b
c
)
a
,
b
,
c
∈
R
a
,
b
,
c
∈
R
a
(
b
+
c
)
=
a
b
+
a
c
(
a
+
b
)
c
=
a
c
+
b
c
(
R
,
+
)
1
∈
R
1
0
1
a
=
a
1
=
a
a
∈
R
R
R
a
b
=
b
a
a
,
b
R
R
a
,
b
∈
R
a
b
=
0
a
=
0
b
=
0
R
R
a
∈
R
a
0
a
−
1
a
−
1
a
=
a
a
−
1
=
1
Z
a
b
=
0
a
b
a
=
0
b
=
0
Z
2
1
/
2
1
−
1
Q
R
C
a
b
Z
n
a
b
(
mod
n
)
Z
12
5
⋅
7
≡
11
(
mod
12
)
Z
n
Z
n
3
⋅
4
≡
0
(
mod
12
)
Z
12
a
R
b
R
a
b
=
0
3
4
Z
12
[
a
,
b
]
f
(
x
)
=
x
2
g
(
x
)
=
cos
x
(
f
+
g
)
(
x
)
=
f
(
x
)
+
g
(
x
)
=
x
2
+
cos
x
(
f
g
)
(
x
)
=
f
(
x
)
g
(
x
)
=
x
2
cos
x
2
×
2
R
A
B
B
A
A
B
=
0
A
B
1
=
(
1
0
0
1
)
,
i
=
(
0
1
−
1
0
)
,
j
=
(
0
i
i
0
)
,
k
=
(
i
0
0
−
i
)
i
2
=
−
1
i
2
=
j
2
=
k
2
=
−
1
i
j
=
k
j
k
=
i
k
i
=
j
j
i
=
−
k
k
j
=
−
i
i
k
=
−
j
H
a
+
b
i
+
c
j
+
d
k
a
,
b
,
c
,
d
H
2
×
2
(
α
β
−
β
¯
α
¯
)
α
=
a
+
d
i
β
=
b
+
c
i
H
1
i
j
k
(
a
1
+
b
1
i
+
c
1
j
+
d
1
k
)
+
(
a
2
+
b
2
i
+
c
2
j
+
d
2
k
)
=
(
a
1
+
a
2
)
+
(
b
1
+
b
2
)
i
+
(
c
1
+
c
2
)
j
+
(
d
1
+
d
2
)
k
(
a
1
+
b
1
i
+
c
1
j
+
d
1
k
)
(
a
2
+
b
2
i
+
c
2
j
+
d
2
k
)
=
α
+
β
i
+
γ
j
+
δ
k
α
=
a
1
a
2
−
b
1
b
2
−
c
1
c
2
−
d
1
d
2
β
=
a
1
b
2
+
a
2
b
1
+
c
1
d
2
−
d
1
c
2
γ
=
a
1
c
2
−
b
1
d
2
+
c
1
a
2
+
d
1
b
2
δ
=
a
1
d
2
+
b
1
c
2
−
c
1
b
2
+
d
1
a
2
H
i
j
k
H
(
a
+
b
i
+
c
j
+
d
k
)
(
a
−
b
i
−
c
j
−
d
k
)
=
a
2
+
b
2
+
c
2
+
d
2
a
b
c
d
a
+
b
i
+
c
j
+
d
k
0
(
a
+
b
i
+
c
j
+
d
k
)
(
a
−
b
i
−
c
j
−
d
k
a
2
+
b
2
+
c
2
+
d
2
)
=
1
R
a
,
b
∈
R
a
0
=
0
a
=
0
a
(
−
b
)
=
(
−
a
)
b
=
−
a
b
(
−
a
)
(
−
b
)
=
a
b
a
0
=
a
(
0
+
0
)
=
a
0
+
a
0
;
a
0
=
0
0
a
=
0
a
b
+
a
(
−
b
)
=
a
(
b
−
b
)
=
a
0
=
0
−
a
b
=
a
(
−
b
)
−
a
b
=
(
−
a
)
b
(
−
a
)
(
−
b
)
=
−
(
a
(
−
b
)
)
=
−
(
−
a
b
)
=
a
b
S
R
S
R
S
R
n
Z
Z
Z
⊂
Q
⊂
R
⊂
C
R
S
R
S
R
S
∅
r
s
∈
S
r
,
s
∈
S
r
−
s
∈
S
r
,
s
∈
S
R
=
M
2
(
R
)
2
×
2
R
T
R
T
=
{
(
a
b
0
c
)
:
a
,
b
,
c
∈
R
}
T
R
A
=
(
a
b
0
c
)
and
B
=
(
a
′
b
′
0
c
′
)
T
A
−
B
T
A
B
=
(
a
a
′
a
b
′
+
b
c
′
0
c
c
′
)
T
R
r
R
r
s
∈
R
r
s
=
0
a
R
a
R
R
i
2
=
−
1
Z
[
i
]
=
{
m
+
n
i
:
m
,
n
∈
Z
}
α
=
a
+
b
i
Z
[
i
]
α
¯
=
a
−
b
i
α
β
=
1
α
¯
β
¯
=
1
β
=
c
+
d
i
1
=
α
β
α
¯
β
¯
=
(
a
2
+
b
2
)
(
c
2
+
d
2
)
a
2
+
b
2
1
−
1
a
+
b
i
=
±
1
a
+
b
i
=
±
i
±
1
±
i
F
=
{
(
1
0
0
1
)
,
(
1
1
1
0
)
,
(
0
1
1
1
)
,
(
0
0
0
0
)
}
Z
2
Q
(
2
)
=
{
a
+
b
2
:
a
,
b
∈
Q
}
a
+
b
2
Q
(
2
)
a
a
2
−
2
b
2
+
−
b
a
2
−
2
b
2
2
D
D
a
∈
D
a
b
=
a
c
b
=
c
D
D
a
b
=
a
c
a
0
a
(
b
−
c
)
=
0
b
−
c
=
0
b
=
c
D
a
b
=
a
c
b
=
c
a
b
=
0
a
0
a
b
=
a
0
b
=
0
a
D
D
∗
D
D
∗
a
∈
D
∗
λ
a
:
D
∗
→
D
∗
λ
a
(
d
)
=
a
d
a
0
d
0
a
d
0
λ
a
d
1
,
d
2
∈
D
∗
a
d
1
=
λ
a
(
d
1
)
=
λ
a
(
d
2
)
=
a
d
2
d
1
=
d
2
D
∗
λ
a
d
∈
D
∗
λ
a
(
d
)
=
a
d
=
1
a
D
d
a
D
n
r
R
r
+
⋯
+
r
n
n
r
R
n
n
r
=
0
r
∈
R
R
0
R
char
R
R
p
Z
p
p
Z
p
Z
p
a
p
a
=
0
Z
p
p
R
1
n
R
n
1
n
n
n
1
=
0
r
∈
R
n
r
=
n
(
1
r
)
=
(
n
1
)
r
=
0
r
=
0
n
n
1
=
0
R
D
D
n
n
0
n
n
=
a
b
1
<
a
<
n
1
<
b
<
n
n
1
=
0
0
=
n
1
=
(
a
b
)
1
=
(
a
1
)
(
b
1
)
D
a
1
=
0
b
1
=
0
D
n
n
R
S
ϕ
:
R
→
S
ϕ
(
a
+
b
)
=
ϕ
(
a
)
+
ϕ
(
b
)
ϕ
(
a
b
)
=
ϕ
(
a
)
ϕ
(
b
)
a
,
b
∈
R
ϕ
:
R
→
S
ϕ
0
ϕ
:
R
→
S
ker
ϕ
=
{
r
∈
R
:
ϕ
(
r
)
=
0
}
n
ϕ
:
Z
→
Z
n
a
↦
a
(
mod
n
)
ϕ
(
a
+
b
)
=
(
a
+
b
)
(
mod
n
)
=
a
(
mod
n
)
+
b
(
mod
n
)
=
ϕ
(
a
)
+
ϕ
(
b
)
ϕ
(
a
b
)
=
a
b
(
mod
n
)
=
a
(
mod
n
)
⋅
b
(
mod
n
)
=
ϕ
(
a
)
ϕ
(
b
)
ϕ
n
Z
C
[
a
,
b
]
[
a
,
b
]
α
∈
[
a
,
b
]
ϕ
α
:
C
[
a
,
b
]
→
R
ϕ
α
(
f
)
=
f
(
α
)
ϕ
α
(
f
+
g
)
=
(
f
+
g
)
(
α
)
=
f
(
α
)
+
g
(
α
)
=
ϕ
α
(
f
)
+
ϕ
α
(
g
)
ϕ
α
(
f
g
)
=
(
f
g
)
(
α
)
=
f
(
α
)
g
(
α
)
=
ϕ
α
(
f
)
ϕ
α
(
g
)
ϕ
α
ϕ
:
R
→
S
R
ϕ
(
R
)
ϕ
(
0
)
=
0
1
R
1
S
R
S
ϕ
ϕ
(
1
R
)
=
1
S
R
ϕ
(
R
)
{
0
}
ϕ
(
R
)
R
I
R
a
I
r
R
a
r
r
a
I
r
I
⊂
I
I
r
⊂
I
r
∈
R
R
{
0
}
R
R
I
R
1
I
r
∈
R
r
1
=
r
∈
I
I
=
R
a
R
⟨
a
⟩
=
{
a
r
:
r
∈
R
}
R
⟨
a
⟩
0
=
a
0
a
=
a
1
⟨
a
⟩
⟨
a
⟩
⟨
a
⟩
a
r
+
a
r
′
=
a
(
r
+
r
′
)
a
r
−
a
r
=
a
(
−
r
)
∈
⟨
a
⟩
a
r
∈
⟨
a
⟩
s
∈
R
s
(
a
r
)
=
a
(
s
r
)
⟨
a
⟩
R
⟨
a
⟩
=
{
a
r
:
r
∈
R
}
Z
{
0
}
⟨
0
⟩
=
{
0
}
I
Z
I
m
n
I
a
I
q
r
a
=
n
q
+
r
0
≤
r
<
n
r
=
a
−
n
q
∈
I
r
0
n
I
a
=
n
q
I
=
⟨
n
⟩
n
Z
n
a
n
Z
b
Z
n
a
b
n
Z
Z
ϕ
:
R
→
S
R
ker
ϕ
R
r
∈
R
a
∈
ker
ϕ
a
r
r
a
ker
ϕ
ϕ
(
a
r
)
=
ϕ
(
a
)
ϕ
(
r
)
=
0
ϕ
(
r
)
=
0
ϕ
(
r
a
)
=
ϕ
(
r
)
ϕ
(
a
)
=
ϕ
(
r
)
0
=
0
r
I
⊂
I
I
r
⊂
I
r
∈
R
r
I
⊂
I
I
r
⊂
I
r
∈
R
I
R
R
/
I
(
r
+
I
)
(
s
+
I
)
=
r
s
+
I
R
/
I
r
+
I
s
+
I
R
/
I
(
r
+
I
)
(
s
+
I
)
=
r
s
+
I
r
′
∈
r
+
I
s
′
∈
s
+
I
r
′
s
′
r
s
+
I
r
′
∈
r
+
I
a
I
r
′
=
r
+
a
b
∈
I
s
′
=
s
+
b
r
′
s
′
=
(
r
+
a
)
(
s
+
b
)
=
r
s
+
a
s
+
r
b
+
a
b
a
s
+
r
b
+
a
b
∈
I
I
r
′
s
′
∈
r
s
+
I
R
/
I
I
R
ϕ
:
R
→
R
/
I
ϕ
(
r
)
=
r
+
I
R
R
/
I
I
ϕ
:
R
→
R
/
I
ϕ
r
s
R
ϕ
(
r
)
ϕ
(
s
)
=
(
r
+
I
)
(
s
+
I
)
=
r
s
+
I
=
ϕ
(
r
s
)
ϕ
:
R
→
R
/
I
ψ
:
R
→
S
ker
ψ
R
ϕ
:
R
→
R
/
ker
ψ
η
:
R
/
ker
ψ
→
ψ
(
R
)
ψ
=
η
ϕ
K
=
ker
ψ
η
:
R
/
K
→
ψ
(
R
)
η
(
r
+
K
)
=
ψ
(
r
)
R
R
/
K
η
(
(
r
+
K
)
(
s
+
K
)
)
=
η
(
r
+
K
)
η
(
s
+
K
)
η
(
(
r
+
K
)
(
s
+
K
)
)
=
η
(
r
s
+
K
)
=
ψ
(
r
s
)
=
ψ
(
r
)
ψ
(
s
)
=
η
(
r
+
K
)
η
(
s
+
K
)
I
R
J
R
I
∩
J
I
I
/
I
∩
J
≅
(
I
+
J
)
/
J
R
I
J
R
J
⊂
I
R
/
I
≅
R
/
J
I
/
J
I
R
S
↦
S
/
I
S
I
R
/
I
R
I
R
/
I
I
R
R
/
I
M
R
R
M
R
R
M
I
M
I
=
R
R
M
R
M
R
R
/
M
M
R
R
R
/
M
1
+
M
R
/
M
R
/
M
a
+
M
R
/
M
a
∉
M
I
{
r
a
+
m
:
r
∈
R
and
m
∈
M
}
I
R
I
0
a
+
0
=
0
I
r
1
a
+
m
1
r
2
a
+
m
2
I
(
r
1
a
+
m
1
)
−
(
r
2
a
+
m
2
)
=
(
r
1
−
r
2
)
a
+
(
m
1
−
m
2
)
I
r
∈
R
r
I
⊂
I
I
I
M
M
I
=
R
I
m
M
b
R
1
=
a
b
+
m
1
+
M
=
a
b
+
M
=
b
a
+
M
=
(
a
+
M
)
(
b
+
M
)
M
R
/
M
R
/
M
0
+
M
=
M
1
+
M
M
R
I
M
I
=
R
a
I
M
a
+
M
b
+
M
R
/
M
(
a
+
M
)
(
b
+
M
)
=
a
b
+
M
=
1
+
M
m
∈
M
a
b
+
m
=
1
1
I
r
1
=
r
∈
I
r
∈
R
I
=
R
p
Z
Z
p
p
Z
Z
/
p
Z
≅
Z
p
P
R
a
b
∈
P
a
∈
P
b
∈
P
P
=
{
0
,
2
,
4
,
6
,
8
,
10
}
Z
12
R
1
1
0
P
R
R
/
P
P
R
R
/
P
a
b
∈
P
a
+
P
b
+
P
R
/
P
(
a
+
P
)
(
b
+
P
)
=
0
+
P
=
P
a
+
P
=
P
b
+
P
=
P
a
P
b
P
P
P
(
a
+
P
)
(
b
+
P
)
=
a
b
+
P
=
0
+
P
=
P
a
b
∈
P
a
∉
P
b
P
b
+
P
=
0
+
P
R
/
P
Z
n
Z
Z
/
n
Z
≅
Z
n
n
Z
p
Z
p
m
n
gcd
(
m
,
n
)
=
1
a
,
b
∈
Z
x
≡
a
(
mod
m
)
x
≡
b
(
mod
n
)
x
1
x
2
x
1
≡
x
2
(
mod
m
n
)
x
≡
a
(
mod
m
)
a
+
k
m
k
∈
Z
k
1
a
+
k
1
m
≡
b
(
mod
n
)
k
1
m
≡
(
b
−
a
)
(
mod
n
)
k
1
m
n
s
t
m
s
+
n
t
=
1
(
b
−
a
)
m
s
=
(
b
−
a
)
−
(
b
−
a
)
n
t
[
(
b
−
a
)
s
]
m
≡
(
b
−
a
)
(
mod
n
)
k
1
=
(
b
−
a
)
s
m
n
c
1
c
2
c
i
≡
a
(
mod
m
)
c
i
≡
b
(
mod
n
)
i
=
1
,
2
c
2
≡
c
1
(
mod
m
)
c
2
≡
c
1
(
mod
n
)
m
n
c
1
−
c
2
c
2
≡
c
1
(
mod
m
n
)
x
≡
3
(
mod
4
)
x
≡
4
(
mod
5
)
s
t
4
s
+
5
t
=
1
s
=
4
t
=
−
3
x
=
a
+
k
1
m
=
3
+
4
k
1
=
3
+
4
[
(
5
−
4
)
4
]
=
19
n
1
,
n
2
,
…
,
n
k
gcd
(
n
i
,
n
j
)
=
1
i
j
a
1
,
…
,
a
k
x
≡
a
1
(
mod
n
1
)
x
≡
a
2
(
mod
n
2
)
⋮
x
≡
a
k
(
mod
n
k
)
n
1
n
2
⋯
n
k
k
=
2
k
x
≡
a
1
(
mod
n
1
)
x
≡
a
2
(
mod
n
2
)
⋮
x
≡
a
k
+
1
(
mod
n
k
+
1
)
k
n
1
⋯
n
k
a
n
1
⋯
n
k
n
k
+
1
x
≡
a
(
mod
n
1
⋯
n
k
)
x
≡
a
k
+
1
(
mod
n
k
+
1
)
n
1
…
n
k
+
1
x
≡
3
(
mod
4
)
x
≡
4
(
mod
5
)
x
≡
1
(
mod
9
)
x
≡
5
(
mod
7
)
19
19
(
mod
20
)
x
≡
19
(
mod
20
)
x
≡
1
(
mod
9
)
x
≡
5
(
mod
7
)
x
≡
19
(
mod
180
)
x
≡
5
(
mod
7
)
19
1260
2
63
−
1
=
9,223,372,036,854,775,807
2
511
−
1
2134
1531
95
97
98
99
2134
≡
44
(
mod
95
)
2134
≡
0
(
mod
97
)
2134
≡
76
(
mod
98
)
2134
≡
55
(
mod
99
)
1531
≡
11
(
mod
95
)
1531
≡
76
(
mod
97
)
1531
≡
61
(
mod
98
)
1531
≡
46
(
mod
99
)
2134
⋅
1531
≡
44
⋅
11
≡
9
(
mod
95
)
2134
⋅
1531
≡
0
⋅
76
≡
0
(
mod
97
)
2134
⋅
1531
≡
76
⋅
61
≡
30
(
mod
98
)
2134
⋅
1531
≡
55
⋅
46
≡
55
(
mod
99
)
2134
⋅
1531
x
≡
9
(
mod
95
)
x
≡
0
(
mod
97
)
x
≡
30
(
mod
98
)
x
≡
55
(
mod
99
)
95
⋅
97
⋅
98
⋅
99
=
89,403,930
x
2134
⋅
1531
=
3,267,154
7
Z
Z
18
Q
(
2
)
=
{
a
+
b
2
:
a
,
b
∈
Q
}
Q
(
2
,
3
)
=
{
a
+
b
2
+
c
3
+
d
6
:
a
,
b
,
c
,
d
∈
Q
}
Z
[
3
]
=
{
a
+
b
3
:
a
,
b
∈
Z
}
R
=
{
a
+
b
3
3
:
a
,
b
∈
Q
}
Z
[
i
]
=
{
a
+
b
i
:
a
,
b
∈
Z
and
i
2
=
−
1
}
Q
(
3
3
)
=
{
a
+
b
3
3
+
c
9
3
:
a
,
b
,
c
∈
Q
}
7
Z
Q
(
2
)
R
R
2
×
2
(
a
b
0
0
)
a
,
b
∈
R
R
S
R
Z
10
Z
12
Z
7
M
2
(
Z
)
2
×
2
Z
M
2
(
Z
2
)
2
×
2
Z
2
{
1
,
3
,
7
,
9
}
{
1
,
2
,
3
,
4
,
5
,
6
}
{
(
1
0
0
1
)
,
(
1
1
0
1
)
,
(
1
0
1
1
)
,
(
0
1
1
0
)
,
(
1
1
1
0
)
,
(
0
1
1
1
)
,
}
Z
18
Z
25
M
2
(
R
)
2
×
2
R
M
2
(
Z
)
2
×
2
Z
Q
{
0
}
{
0
,
9
}
{
0
,
6
,
12
}
{
0
,
3
,
6
,
9
,
12
,
15
}
{
0
,
2
,
4
,
6
,
8
,
10
,
12
,
14
,
16
}
R
I
R
/
I
R
=
Z
I
=
6
Z
R
=
Z
12
I
=
{
0
,
3
,
6
,
9
}
ϕ
:
Z
/
6
Z
→
Z
/
15
Z
R
C
ϕ
:
C
→
R
ϕ
(
i
)
=
a
Q
(
2
)
=
{
a
+
b
2
:
a
,
b
∈
Q
}
Q
(
3
)
=
{
a
+
b
3
:
a
,
b
∈
Q
}
ϕ
:
Q
(
2
)
→
Q
(
3
)
ϕ
(
2
)
=
a
F
=
{
(
1
0
0
1
)
,
(
1
1
1
0
)
,
(
0
1
1
1
)
,
(
0
0
0
0
)
}
Z
2
ϕ
:
C
→
M
2
(
R
)
ϕ
(
a
+
b
i
)
=
(
a
b
−
b
a
)
ϕ
C
M
2
(
R
)
Z
[
i
]
Z
[
3
i
]
=
{
a
+
b
3
i
:
a
,
b
∈
Z
}
x
≡
2
(
mod
5
)
x
≡
6
(
mod
11
)
x
≡
3
(
mod
7
)
x
≡
0
(
mod
8
)
x
≡
5
(
mod
15
)
x
≡
2
(
mod
4
)
x
≡
4
(
mod
7
)
x
≡
7
(
mod
9
)
x
≡
5
(
mod
11
)
x
≡
3
(
mod
5
)
x
≡
0
(
mod
8
)
x
≡
1
(
mod
11
)
x
≡
5
(
mod
13
)
x
≡
17
(
mod
55
)
x
≡
214
(
mod
2772
)
2234
+
4121
95
97
98
99
2134
⋅
1531
98
99
R
R
{
0
}
R
I
{
0
}
1
∈
I
a
R
(
−
1
)
a
=
−
a
ϕ
:
R
→
S
R
ϕ
(
R
)
ϕ
(
0
)
=
0
1
R
1
S
R
S
ϕ
ϕ
(
1
R
)
=
1
S
R
ϕ
(
R
)
0
ϕ
(
R
)
ϕ
(
a
)
ϕ
(
b
)
=
ϕ
(
a
b
)
=
ϕ
(
b
a
)
=
ϕ
(
b
)
ϕ
(
a
)
R
/
I
I
R
J
R
I
∩
J
I
I
/
I
∩
J
≅
I
+
J
/
J
R
I
J
R
J
⊂
I
R
/
I
≅
R
/
J
I
/
J
I
R
S
→
S
/
I
S
I
R
/
I
R
R
/
I
R
S
R
S
R
S
∅
r
s
∈
S
r
,
s
∈
S
r
−
s
∈
S
r
,
s
∈
S
R
{
R
α
}
⋂
R
α
R
{
I
α
}
α
∈
A
R
⋂
α
∈
A
I
α
R
I
1
I
2
R
I
1
∪
I
2
R
R
{
0
}
R
R
a
∈
R
a
0
a
R
b
∈
R
a
b
=
1
R
a
R
a
n
=
0
n
R
R
a
∈
R
a
2
=
a
(
a
+
b
)
2
(
−
a
b
)
2
R
a
3
=
a
a
∈
R
R
R
1
R
S
R
1
S
1
R
=
1
S
R
1
=
0
R
=
{
0
}
R
R
Z
(
R
)
=
{
a
∈
R
:
a
r
=
r
a
for all
r
∈
R
}
Z
(
R
)
R
p
Z
(
p
)
=
{
a
/
b
:
a
,
b
∈
Z
and
gcd
(
b
,
p
)
=
1
}
Z
(
p
)
p
p
a
/
b
,
c
/
d
∈
Z
(
p
)
a
/
b
+
c
/
d
=
(
a
d
+
b
c
)
/
b
d
(
a
/
b
)
⋅
(
c
/
d
)
=
(
a
c
)
/
(
b
d
)
Z
(
p
)
gcd
(
b
d
,
p
)
=
1
Z
p
R
u
R
i
u
:
R
→
R
r
↦
u
r
u
−
1
i
u
R
R
R
R
Inn
(
R
)
R
Aut
(
R
)
Inn
(
R
)
Aut
(
R
)
U
(
R
)
R
ϕ
:
U
(
R
)
→
Inn
(
R
)
u
↦
i
u
ϕ
Aut
(
Z
)
Inn
(
Z
)
U
(
Z
)
R
S
R
×
S
(
r
,
s
)
+
(
r
′
,
s
′
)
=
(
r
+
r
′
,
s
+
s
′
)
(
r
,
s
)
(
r
′
,
s
′
)
=
(
r
r
′
,
s
s
′
)
x
x
2
=
x
0
1
x
x
2
=
x
x
0
R
x
=
1
M
2
(
R
)
gcd
(
a
,
n
)
=
d
gcd
(
b
,
d
)
1
a
x
≡
b
(
mod
n
)
R
I
J
R
I
+
J
=
R
r
s
R
x
≡
r
(
mod
I
)
x
≡
s
(
mod
J
)
I
∩
J
I
J
R
I
+
J
=
R
R
/
(
I
∩
J
)
≅
R
/
I
×
R
/
J
Z
n
Z
n
n
n
Q
R
C
x
2
−
n
=
0
Q
[
n
]
x
n
−
1
=
0
Q
¯
p
Z
p
7
x
2
−
7
7
−
7
x
2
−
7
x
2
−
7
r
2
=
n
s
2
=
m
t
=
r
s
=
−
s
r
Z
Z
4
y
y
¯
Z
11
Z
12
1
Z
⟨
4
⟩
{
a
⋅
3
+
b
⋅
5
∣
a
,
b
∈
Z
}
Z
F
F
F
F
Z
3
P
z
Z
7
K
z
2
+
z
+
3
H
P
K
H
p
(
x
)
=
x
3
−
3
x
+
2
q
(
x
)
=
3
x
2
−
6
x
+
5
p
(
x
)
+
q
(
x
)
p
(
x
)
q
(
x
)
(
p
+
q
)
(
x
)
=
p
(
x
)
+
q
(
x
)
=
(
x
3
−
3
x
+
2
)
+
(
3
x
2
−
6
x
+
5
)
=
x
3
+
3
x
2
−
9
x
+
7
(
p
q
)
(
x
)
=
p
(
x
)
q
(
x
)
=
(
x
3
−
3
x
+
2
)
(
3
x
2
−
6
x
+
5
)
=
3
x
5
−
6
x
4
−
4
x
3
+
24
x
2
−
27
x
+
10
R
f
(
x
)
=
∑
i
=
0
n
a
i
x
i
=
a
0
+
a
1
x
+
a
2
x
2
+
⋯
+
a
n
x
n
a
i
∈
R
a
n
0
R
x
a
0
,
a
1
,
…
,
a
n
f
a
n
n
a
n
0
f
n
deg
f
(
x
)
=
n
n
f
=
0
f
−
∞
R
R
[
x
]
R
p
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
q
(
x
)
=
b
0
+
b
1
x
+
⋯
+
b
m
x
m
p
(
x
)
=
q
(
x
)
a
i
=
b
i
i
≥
0
p
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
q
(
x
)
=
b
0
+
b
1
x
+
⋯
+
b
m
x
m
p
(
x
)
q
(
x
)
p
(
x
)
+
q
(
x
)
=
c
0
+
c
1
x
+
⋯
+
c
k
x
k
c
i
=
a
i
+
b
i
i
p
(
x
)
q
(
x
)
p
(
x
)
q
(
x
)
=
c
0
+
c
1
x
+
⋯
+
c
m
+
n
x
m
+
n
c
i
=
∑
k
=
0
i
a
k
b
i
−
k
=
a
0
b
i
+
a
1
b
i
−
1
+
⋯
+
a
i
−
1
b
1
+
a
i
b
0
i
p
(
x
)
=
3
+
0
x
+
0
x
2
+
2
x
3
+
0
x
4
q
(
x
)
=
2
+
0
x
−
x
2
+
0
x
3
+
4
x
4
Z
[
x
]
p
(
x
)
=
3
+
2
x
3
q
(
x
)
=
2
−
x
2
+
4
x
4
p
(
x
)
+
q
(
x
)
=
5
−
x
2
+
2
x
3
+
4
x
4
p
(
x
)
q
(
x
)
=
(
3
+
2
x
3
)
(
2
−
x
2
+
4
x
4
)
=
6
−
3
x
2
+
4
x
3
+
12
x
4
−
2
x
5
+
8
x
7
c
i
p
(
x
)
=
3
+
3
x
3
and
q
(
x
)
=
4
+
4
x
2
+
4
x
4
Z
12
[
x
]
p
(
x
)
q
(
x
)
7
+
4
x
2
+
3
x
3
+
4
x
4
R
[
x
]
R
R
R
[
x
]
R
[
x
]
f
(
x
)
=
0
p
(
x
)
=
∑
i
=
0
n
a
i
x
i
p
(
x
)
−
p
(
x
)
=
∑
i
=
0
n
(
−
a
i
)
x
i
=
−
∑
i
=
0
n
a
i
x
i
R
p
(
x
)
=
∑
i
=
0
m
a
i
x
i
,
q
(
x
)
=
∑
i
=
0
n
b
i
x
i
,
r
(
x
)
=
∑
i
=
0
p
c
i
x
i
[
p
(
x
)
q
(
x
)
]
r
(
x
)
=
[
(
∑
i
=
0
m
a
i
x
i
)
(
∑
i
=
0
n
b
i
x
i
)
]
(
∑
i
=
0
p
c
i
x
i
)
=
[
∑
i
=
0
m
+
n
(
∑
j
=
0
i
a
j
b
i
−
j
)
x
i
]
(
∑
i
=
0
p
c
i
x
i
)
=
∑
i
=
0
m
+
n
+
p
[
∑
j
=
0
i
(
∑
k
=
0
j
a
k
b
j
−
k
)
c
i
−
j
]
x
i
=
∑
i
=
0
m
+
n
+
p
(
∑
j
+
k
+
l
=
i
a
j
b
k
c
l
)
x
i
=
∑
i
=
0
m
+
n
+
p
[
∑
j
=
0
i
a
j
(
∑
k
=
0
i
−
j
b
k
c
i
−
j
−
k
)
]
x
i
=
(
∑
i
=
0
m
a
i
x
i
)
[
∑
i
=
0
n
+
p
(
∑
j
=
0
i
b
j
c
i
−
j
)
x
i
]
=
(
∑
i
=
0
m
a
i
x
i
)
[
(
∑
i
=
0
n
b
i
x
i
)
(
∑
i
=
0
p
c
i
x
i
)
]
=
p
(
x
)
[
q
(
x
)
r
(
x
)
]
p
(
x
)
q
(
x
)
R
[
x
]
R
deg
p
(
x
)
+
deg
q
(
x
)
=
deg
(
p
(
x
)
q
(
x
)
)
R
[
x
]
p
(
x
)
=
a
m
x
m
+
⋯
+
a
1
x
+
a
0
q
(
x
)
=
b
n
x
n
+
⋯
+
b
1
x
+
b
0
a
m
0
b
n
0
p
(
x
)
q
(
x
)
m
n
p
(
x
)
q
(
x
)
a
m
b
n
x
m
+
n
R
p
(
x
)
q
(
x
)
m
+
n
p
(
x
)
q
(
x
)
0
p
(
x
)
0
q
(
x
)
0
p
(
x
)
q
(
x
)
0
R
[
x
]
x
2
−
3
x
y
+
2
y
3
R
x
y
(
R
[
x
]
)
[
y
]
(
R
[
x
]
)
[
y
]
≅
R
(
[
y
]
)
[
x
]
R
[
x
,
y
]
R
[
x
,
y
]
x
y
R
n
n
R
R
[
x
1
,
x
2
,
…
,
x
n
]
n
R
α
∈
R
ϕ
α
:
R
[
x
]
→
R
α
ϕ
α
(
p
(
x
)
)
=
p
(
α
)
=
a
n
α
n
+
⋯
+
a
1
α
+
a
0
p
(
x
)
=
a
n
x
n
+
⋯
+
a
1
x
+
a
0
p
(
x
)
=
∑
i
=
0
n
a
i
x
i
q
(
x
)
=
∑
i
=
0
m
b
i
x
i
ϕ
α
(
p
(
x
)
+
q
(
x
)
)
=
ϕ
α
(
p
(
x
)
)
+
ϕ
α
(
q
(
x
)
)
ϕ
α
ϕ
α
(
p
(
x
)
)
ϕ
α
(
q
(
x
)
)
=
p
(
α
)
q
(
α
)
=
(
∑
i
=
0
n
a
i
α
i
)
(
∑
i
=
0
m
b
i
α
i
)
=
∑
i
=
0
m
+
n
(
∑
k
=
0
i
a
k
b
i
−
k
)
α
i
=
ϕ
α
(
p
(
x
)
q
(
x
)
)
ϕ
α
:
R
[
x
]
→
R
α
a
b
b
>
0
q
r
a
=
b
q
+
r
0
≤
r
<
b
q
r
f
(
x
)
g
(
x
)
F
[
x
]
F
g
(
x
)
q
(
x
)
,
r
(
x
)
∈
F
[
x
]
f
(
x
)
=
g
(
x
)
q
(
x
)
+
r
(
x
)
deg
r
(
x
)
<
deg
g
(
x
)
r
(
x
)
q
(
x
)
r
(
x
)
f
(
x
)
0
=
0
⋅
g
(
x
)
+
0
;
q
r
f
(
x
)
deg
f
(
x
)
=
n
deg
g
(
x
)
=
m
m
>
n
q
(
x
)
=
0
r
(
x
)
=
f
(
x
)
m
≤
n
n
f
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
1
x
+
a
0
g
(
x
)
=
b
m
x
m
+
b
m
−
1
x
m
−
1
+
⋯
+
b
1
x
+
b
0
f
′
(
x
)
=
f
(
x
)
−
a
n
b
m
x
n
−
m
g
(
x
)
n
q
′
(
x
)
r
(
x
)
f
′
(
x
)
=
q
′
(
x
)
g
(
x
)
+
r
(
x
)
r
(
x
)
=
0
r
(
x
)
g
(
x
)
q
(
x
)
=
q
′
(
x
)
+
a
n
b
m
x
n
−
m
f
(
x
)
=
g
(
x
)
q
(
x
)
+
r
(
x
)
r
(
x
)
deg
r
(
x
)
<
deg
g
(
x
)
q
(
x
)
r
(
x
)
q
1
(
x
)
r
1
(
x
)
f
(
x
)
=
g
(
x
)
q
1
(
x
)
+
r
1
(
x
)
deg
r
1
(
x
)
<
deg
g
(
x
)
r
1
(
x
)
=
0
f
(
x
)
=
g
(
x
)
q
(
x
)
+
r
(
x
)
=
g
(
x
)
q
1
(
x
)
+
r
1
(
x
)
g
(
x
)
[
q
(
x
)
−
q
1
(
x
)
]
=
r
1
(
x
)
−
r
(
x
)
q
(
x
)
−
q
1
(
x
)
deg
(
g
(
x
)
[
q
(
x
)
−
q
1
(
x
)
]
)
=
deg
(
r
1
(
x
)
−
r
(
x
)
)
≥
deg
g
(
x
)
r
(
x
)
r
1
(
x
)
g
(
x
)
r
(
x
)
=
r
1
(
x
)
q
(
x
)
=
q
1
(
x
)
x
3
−
x
2
+
2
x
−
3
x
−
2
x
2
+
x
+
4
x
−
2
x
3
−
x
2
+
2
x
−
3
x
3
−
2
x
2
x
2
+
2
x
−
3
x
2
−
2
x
4
x
−
3
4
x
−
8
5
x
3
−
x
2
+
2
x
−
3
=
(
x
−
2
)
(
x
2
+
x
+
4
)
+
5
p
(
x
)
F
[
x
]
α
∈
F
α
p
(
x
)
p
(
x
)
ϕ
α
α
p
(
x
)
p
(
α
)
=
0
F
α
∈
F
p
(
x
)
∈
F
[
x
]
x
−
α
p
(
x
)
F
[
x
]
α
∈
F
p
(
α
)
=
0
q
(
x
)
r
(
x
)
p
(
x
)
=
(
x
−
α
)
q
(
x
)
+
r
(
x
)
r
(
x
)
x
−
α
r
(
x
)
r
(
x
)
=
a
a
∈
F
p
(
x
)
=
(
x
−
α
)
q
(
x
)
+
a
0
=
p
(
α
)
=
0
⋅
q
(
α
)
+
a
=
a
;
p
(
x
)
=
(
x
−
α
)
q
(
x
)
x
−
α
p
(
x
)
x
−
α
p
(
x
)
p
(
x
)
=
(
x
−
α
)
q
(
x
)
p
(
α
)
=
0
⋅
q
(
α
)
=
0
F
p
(
x
)
n
F
[
x
]
n
F
p
(
x
)
deg
p
(
x
)
=
0
p
(
x
)
deg
p
(
x
)
=
1
p
(
x
)
=
a
x
+
b
a
b
F
α
1
α
2
p
(
x
)
a
α
1
+
b
=
a
α
2
+
b
α
1
=
α
2
deg
p
(
x
)
>
1
p
(
x
)
F
α
p
(
x
)
p
(
x
)
=
(
x
−
α
)
q
(
x
)
q
(
x
)
∈
F
[
x
]
q
(
x
)
n
−
1
β
p
(
x
)
α
p
(
β
)
=
(
β
−
α
)
q
(
β
)
=
0
α
β
F
q
(
β
)
=
0
q
(
x
)
n
−
1
F
α
p
(
x
)
n
F
F
d
(
x
)
p
(
x
)
,
q
(
x
)
∈
F
[
x
]
d
(
x
)
p
(
x
)
q
(
x
)
d
′
(
x
)
p
(
x
)
q
(
x
)
d
′
(
x
)
∣
d
(
x
)
d
(
x
)
=
gcd
(
p
(
x
)
,
q
(
x
)
)
p
(
x
)
q
(
x
)
gcd
(
p
(
x
)
,
q
(
x
)
)
=
1
F
d
(
x
)
p
(
x
)
q
(
x
)
F
[
x
]
r
(
x
)
s
(
x
)
d
(
x
)
=
r
(
x
)
p
(
x
)
+
s
(
x
)
q
(
x
)
d
(
x
)
S
=
{
f
(
x
)
p
(
x
)
+
g
(
x
)
q
(
x
)
:
f
(
x
)
,
g
(
x
)
∈
F
[
x
]
}
d
(
x
)
=
r
(
x
)
p
(
x
)
+
s
(
x
)
q
(
x
)
r
(
x
)
s
(
x
)
F
[
x
]
d
(
x
)
p
(
x
)
q
(
x
)
d
(
x
)
p
(
x
)
a
(
x
)
b
(
x
)
p
(
x
)
=
a
(
x
)
d
(
x
)
+
b
(
x
)
b
(
x
)
deg
b
(
x
)
<
deg
d
(
x
)
b
(
x
)
=
p
(
x
)
−
a
(
x
)
d
(
x
)
=
p
(
x
)
−
a
(
x
)
(
r
(
x
)
p
(
x
)
+
s
(
x
)
q
(
x
)
)
=
p
(
x
)
−
a
(
x
)
r
(
x
)
p
(
x
)
−
a
(
x
)
s
(
x
)
q
(
x
)
=
p
(
x
)
(
1
−
a
(
x
)
r
(
x
)
)
+
q
(
x
)
(
−
a
(
x
)
s
(
x
)
)
p
(
x
)
q
(
x
)
S
b
(
x
)
d
(
x
)
d
(
x
)
p
(
x
)
d
(
x
)
q
(
x
)
d
(
x
)
p
(
x
)
q
(
x
)
d
(
x
)
p
(
x
)
q
(
x
)
d
′
(
x
)
p
(
x
)
q
(
x
)
d
′
(
x
)
∣
d
(
x
)
d
′
(
x
)
p
(
x
)
q
(
x
)
u
(
x
)
v
(
x
)
p
(
x
)
=
u
(
x
)
d
′
(
x
)
q
(
x
)
=
v
(
x
)
d
′
(
x
)
d
(
x
)
=
r
(
x
)
p
(
x
)
+
s
(
x
)
q
(
x
)
=
r
(
x
)
u
(
x
)
d
′
(
x
)
+
s
(
x
)
v
(
x
)
d
′
(
x
)
=
d
′
(
x
)
[
r
(
x
)
u
(
x
)
+
s
(
x
)
v
(
x
)
]
d
′
(
x
)
∣
d
(
x
)
d
(
x
)
p
(
x
)
q
(
x
)
p
(
x
)
q
(
x
)
d
′
(
x
)
p
(
x
)
q
(
x
)
u
(
x
)
v
(
x
)
F
[
x
]
d
(
x
)
=
d
′
(
x
)
[
r
(
x
)
u
(
x
)
+
s
(
x
)
v
(
x
)
]
deg
d
(
x
)
=
deg
d
′
(
x
)
+
deg
[
r
(
x
)
u
(
x
)
+
s
(
x
)
v
(
x
)
]
d
(
x
)
d
′
(
x
)
deg
d
(
x
)
=
deg
d
′
(
x
)
d
(
x
)
d
′
(
x
)
d
(
x
)
=
d
′
(
x
)
f
(
x
)
∈
F
[
x
]
F
f
(
x
)
g
(
x
)
h
(
x
)
F
[
x
]
g
(
x
)
h
(
x
)
f
(
x
)
x
2
−
2
∈
Q
[
x
]
x
2
+
1
p
(
x
)
=
x
3
+
x
2
+
2
Z
3
[
x
]
Z
3
[
x
]
x
−
a
a
Z
3
[
x
]
p
(
a
)
=
0
p
(
0
)
=
2
p
(
1
)
=
1
p
(
2
)
=
2
p
(
x
)
Z
3
p
(
x
)
∈
Q
[
x
]
p
(
x
)
=
r
s
(
a
0
+
a
1
x
+
⋯
+
a
n
x
n
)
r
,
s
,
a
0
,
…
,
a
n
a
i
r
s
p
(
x
)
=
b
0
c
0
+
b
1
c
1
x
+
⋯
+
b
n
c
n
x
n
b
i
c
i
p
(
x
)
p
(
x
)
=
1
c
0
⋯
c
n
(
d
0
+
d
1
x
+
⋯
+
d
n
x
n
)
d
0
,
…
,
d
n
d
d
0
,
…
,
d
n
p
(
x
)
=
d
c
0
⋯
c
n
(
a
0
+
a
1
x
+
⋯
+
a
n
x
n
)
d
i
=
d
a
i
a
i
d
/
(
c
0
⋯
c
n
)
p
(
x
)
=
r
s
(
a
0
+
a
1
x
+
⋯
+
a
n
x
n
)
gcd
(
r
,
s
)
=
1
p
(
x
)
∈
Z
[
x
]
p
(
x
)
α
(
x
)
β
(
x
)
Q
[
x
]
α
(
x
)
β
(
x
)
p
(
x
)
p
(
x
)
=
a
(
x
)
b
(
x
)
a
(
x
)
b
(
x
)
Z
[
x
]
deg
α
(
x
)
=
deg
a
(
x
)
deg
β
(
x
)
=
deg
b
(
x
)
α
(
x
)
=
c
1
d
1
(
a
0
+
a
1
x
+
⋯
+
a
m
x
m
)
=
c
1
d
1
α
1
(
x
)
β
(
x
)
=
c
2
d
2
(
b
0
+
b
1
x
+
⋯
+
b
n
x
n
)
=
c
2
d
2
β
1
(
x
)
a
i
b
i
p
(
x
)
=
α
(
x
)
β
(
x
)
=
c
1
c
2
d
1
d
2
α
1
(
x
)
β
1
(
x
)
=
c
d
α
1
(
x
)
β
1
(
x
)
c
/
d
c
1
/
d
1
c
2
/
d
2
d
p
(
x
)
=
c
α
1
(
x
)
β
1
(
x
)
d
=
1
c
a
m
b
n
=
1
p
(
x
)
c
=
1
c
=
−
1
c
=
1
a
m
=
b
n
=
1
a
m
=
b
n
=
−
1
p
(
x
)
=
α
1
(
x
)
β
1
(
x
)
α
1
(
x
)
β
1
(
x
)
deg
α
(
x
)
=
deg
α
1
(
x
)
deg
β
(
x
)
=
deg
β
1
(
x
)
a
(
x
)
=
−
α
1
(
x
)
b
(
x
)
=
−
β
1
(
x
)
p
(
x
)
=
(
−
α
1
(
x
)
)
(
−
β
1
(
x
)
)
=
a
(
x
)
b
(
x
)
c
=
−
1
d
1
gcd
(
c
,
d
)
=
1
p
p
∣
d
p
∤
c
α
1
(
x
)
a
i
p
∤
a
i
b
j
β
1
(
x
)
p
∤
b
j
α
1
′
(
x
)
β
1
′
(
x
)
Z
p
[
x
]
α
1
(
x
)
β
1
(
x
)
p
p
∣
d
α
1
′
(
x
)
β
1
′
(
x
)
=
0
Z
p
[
x
]
α
1
′
(
x
)
β
1
′
(
x
)
Z
p
[
x
]
d
=
1
p
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
0
Z
a
0
0
p
(
x
)
Q
p
(
x
)
α
Z
α
a
0
p
(
x
)
a
∈
Q
p
(
x
)
x
−
a
p
(
x
)
Z
[
x
]
α
∈
Z
p
(
x
)
=
(
x
−
α
)
(
x
n
−
1
+
⋯
−
a
0
/
α
)
a
0
/
α
∈
Z
α
∣
a
0
p
(
x
)
=
x
4
−
2
x
3
+
x
+
1
p
(
x
)
Q
[
x
]
p
(
x
)
p
(
x
)
p
(
x
)
=
(
x
−
α
)
q
(
x
)
q
(
x
)
p
(
x
)
p
(
x
)
Q
[
x
]
Z
±
1
p
(
1
)
=
1
p
(
−
1
)
=
3
p
(
x
)
p
(
x
)
p
(
x
)
=
(
x
2
+
a
x
+
b
)
(
x
2
+
c
x
+
d
)
=
x
4
+
(
a
+
c
)
x
3
+
(
a
c
+
b
+
d
)
x
2
+
(
a
d
+
b
c
)
x
+
b
d
Z
[
x
]
a
+
c
=
−
2
a
c
+
b
+
d
=
0
a
d
+
b
c
=
1
b
d
=
1
b
d
=
1
b
=
d
=
1
b
=
d
=
−
1
b
=
d
a
d
+
b
c
=
b
(
a
+
c
)
=
1
a
+
c
=
−
2
−
2
b
=
1
b
p
(
x
)
Q
p
f
(
x
)
=
a
n
x
n
+
⋯
+
a
0
∈
Z
[
x
]
p
∣
a
i
i
=
0
,
1
,
…
,
n
−
1
p
∤
a
n
p
2
∤
a
0
f
(
x
)
Q
f
(
x
)
Z
[
x
]
f
(
x
)
=
(
b
r
x
r
+
⋯
+
b
0
)
(
c
s
x
s
+
⋯
+
c
0
)
Z
[
x
]
b
r
c
s
r
,
s
<
n
p
2
a
0
=
b
0
c
0
b
0
c
0
p
p
∤
b
0
p
∣
c
0
p
∤
a
n
a
n
=
b
r
c
s
b
r
c
s
p
m
k
p
∤
c
k
a
m
=
b
0
c
m
+
b
1
c
m
−
1
+
⋯
+
b
m
c
0
p
p
b
0
c
m
m
=
n
a
i
p
m
<
n
f
(
x
)
f
(
x
)
=
16
x
5
−
9
x
4
+
3
x
2
+
6
x
−
21
Q
p
=
3
Q
Q
[
x
]
F
[
x
]
F
F
[
x
]
⟨
p
(
x
)
⟩
p
(
x
)
⟨
p
(
x
)
⟩
=
{
p
(
x
)
q
(
x
)
:
q
(
x
)
∈
F
[
x
]
}
x
2
F
[
x
]
⟨
x
2
⟩
1
F
F
[
x
]
I
F
[
x
]
I
I
F
[
x
]
p
(
x
)
∈
I
deg
p
(
x
)
=
0
p
(
x
)
I
F
[
x
]
⟨
1
⟩
=
I
=
F
[
x
]
I
deg
p
(
x
)
≥
1
f
(
x
)
I
q
(
x
)
r
(
x
)
F
[
x
]
f
(
x
)
=
p
(
x
)
q
(
x
)
+
r
(
x
)
deg
r
(
x
)
<
deg
p
(
x
)
f
(
x
)
,
p
(
x
)
∈
I
I
r
(
x
)
=
f
(
x
)
−
p
(
x
)
q
(
x
)
I
p
(
x
)
r
(
x
)
f
(
x
)
I
p
(
x
)
q
(
x
)
q
(
x
)
∈
F
[
x
]
I
=
⟨
p
(
x
)
⟩
F
[
x
,
y
]
F
[
x
,
y
]
x
y
F
[
x
,
y
]
x
y
F
p
(
x
)
∈
F
[
x
]
p
(
x
)
p
(
x
)
p
(
x
)
F
[
x
]
⟨
p
(
x
)
⟩
F
[
x
]
F
[
x
]
p
(
x
)
p
(
x
)
p
(
x
)
=
f
(
x
)
g
(
x
)
⟨
p
(
x
)
⟩
f
(
x
)
⟨
p
(
x
)
⟩
p
(
x
)
⟨
p
(
x
)
⟩
⊂
⟨
f
(
x
)
⟩
⟨
p
(
x
)
⟩
p
(
x
)
F
[
x
]
I
F
[
x
]
⟨
p
(
x
)
⟩
I
I
=
⟨
f
(
x
)
⟩
f
(
x
)
∈
F
[
x
]
p
(
x
)
∈
I
p
(
x
)
=
f
(
x
)
g
(
x
)
g
(
x
)
∈
F
[
x
]
p
(
x
)
f
(
x
)
g
(
x
)
f
(
x
)
I
=
F
[
x
]
g
(
x
)
f
(
x
)
I
I
=
⟨
p
(
x
)
⟩
F
[
x
]
⟨
p
(
x
)
⟩
a
x
2
+
b
x
+
c
=
0
a
x
3
+
b
x
2
+
c
x
+
d
=
0
a
x
3
+
c
x
+
d
=
0
a
x
3
+
b
x
2
+
c
x
+
d
=
0
a
x
4
+
b
x
3
+
c
x
2
+
d
x
+
e
=
0
p
(
x
)
n
p
(
x
)
8
x
5
−
18
x
4
+
20
x
3
−
25
x
2
+
20
4
x
2
−
x
−
2
3
Z
2
[
x
]
(
5
x
2
+
3
x
−
4
)
+
(
4
x
2
−
x
+
9
)
Z
12
(
5
x
2
+
3
x
−
4
)
(
4
x
2
−
x
+
9
)
Z
12
(
7
x
3
+
3
x
2
−
x
)
+
(
6
x
2
−
8
x
+
4
)
Z
9
(
3
x
2
+
2
x
−
4
)
+
(
4
x
2
+
2
)
Z
5
(
3
x
2
+
2
x
−
4
)
(
4
x
2
+
2
)
Z
5
(
5
x
2
+
3
x
−
2
)
2
Z
12
9
x
2
+
2
x
+
5
8
x
4
+
7
x
3
+
2
x
2
+
7
x
q
(
x
)
r
(
x
)
a
(
x
)
=
q
(
x
)
b
(
x
)
+
r
(
x
)
deg
r
(
x
)
<
deg
b
(
x
)
a
(
x
)
=
5
x
3
+
6
x
2
−
3
x
+
4
b
(
x
)
=
x
−
2
Z
7
[
x
]
a
(
x
)
=
6
x
4
−
2
x
3
+
x
2
−
3
x
+
1
b
(
x
)
=
x
2
+
x
−
2
Z
7
[
x
]
a
(
x
)
=
4
x
5
−
x
3
+
x
2
+
4
b
(
x
)
=
x
3
−
2
Z
5
[
x
]
a
(
x
)
=
x
5
+
x
3
−
x
2
−
x
b
(
x
)
=
x
3
+
x
Z
2
[
x
]
5
x
3
+
6
x
2
−
3
x
+
4
=
(
5
x
2
+
2
x
+
1
)
(
x
−
2
)
+
6
4
x
5
−
x
3
+
x
2
+
4
=
(
4
x
2
+
4
)
(
x
3
+
3
)
+
4
x
2
+
2
p
(
x
)
q
(
x
)
d
(
x
)
=
gcd
(
p
(
x
)
,
q
(
x
)
)
a
(
x
)
b
(
x
)
a
(
x
)
p
(
x
)
+
b
(
x
)
q
(
x
)
=
d
(
x
)
p
(
x
)
=
x
3
−
6
x
2
+
14
x
−
15
q
(
x
)
=
x
3
−
8
x
2
+
21
x
−
18
p
(
x
)
,
q
(
x
)
∈
Q
[
x
]
p
(
x
)
=
x
3
+
x
2
−
x
+
1
q
(
x
)
=
x
3
+
x
−
1
p
(
x
)
,
q
(
x
)
∈
Z
2
[
x
]
p
(
x
)
=
x
3
+
x
2
−
4
x
+
4
q
(
x
)
=
x
3
+
3
x
−
2
p
(
x
)
,
q
(
x
)
∈
Z
5
[
x
]
p
(
x
)
=
x
3
−
2
x
+
4
q
(
x
)
=
4
x
3
+
x
+
3
p
(
x
)
,
q
(
x
)
∈
Q
[
x
]
5
x
3
+
4
x
2
−
x
+
9
Z
12
3
x
3
−
4
x
2
−
x
+
4
Z
5
5
x
4
+
2
x
2
−
3
Z
7
x
3
+
x
+
1
Z
2
Z
12
3
4
Z
[
x
]
p
(
x
)
Z
4
[
x
]
deg
p
(
x
)
>
1
(
2
x
+
1
)
Q
[
x
]
x
4
−
2
x
3
+
2
x
2
+
x
+
4
x
4
−
5
x
3
+
3
x
−
2
3
x
5
−
4
x
3
−
6
x
2
+
6
5
x
5
−
6
x
4
−
3
x
2
+
9
x
−
15
2
3
Z
2
[
x
]
x
2
+
x
+
8
Z
10
[
x
]
x
2
+
x
+
8
=
(
x
+
2
)
(
x
+
9
)
p
(
x
)
Z
6
[
x
]
n
n
F
F
[
x
1
,
…
,
x
n
]
Z
[
x
]
Z
x
p
+
a
a
∈
Z
p
p
f
(
x
)
F
[
x
]
F
f
(
x
)
∣
p
(
x
)
q
(
x
)
f
(
x
)
∣
p
(
x
)
f
(
x
)
∣
q
(
x
)
R
S
R
[
x
]
≅
S
[
x
]
ϕ
:
R
→
S
ϕ
¯
:
R
[
x
]
→
S
[
x
]
ϕ
¯
(
a
0
+
a
1
x
+
⋯
+
a
n
x
n
)
=
ϕ
(
a
0
)
+
ϕ
(
a
1
)
x
+
⋯
+
ϕ
(
a
n
)
x
n
F
a
∈
F
p
(
x
)
∈
F
[
x
]
p
(
a
)
p
(
x
)
x
−
a
p
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
0
∈
Z
[
x
]
a
n
0
p
(
r
/
s
)
=
0
gcd
(
r
,
s
)
=
1
r
∣
a
0
s
∣
a
n
Q
∗
Q
∗
(
Z
[
x
]
,
+
)
Φ
n
(
x
)
=
x
n
−
1
x
−
1
=
x
n
−
1
+
x
n
−
2
+
⋯
+
x
+
1
Φ
p
(
x
)
Q
p
Φ
n
(
x
)
=
x
n
−
1
x
−
1
=
x
n
−
1
+
x
n
−
2
+
⋯
+
x
+
1
Φ
p
(
x
)
Q
p
F
F
[
x
]
R
R
[
x
]
R
R
[
x
]
x
p
−
x
p
Z
p
p
x
p
−
x
=
x
(
x
−
1
)
(
x
−
2
)
⋯
(
x
−
(
p
−
1
)
)
F
f
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
F
[
x
]
f
′
(
x
)
=
a
1
+
2
a
2
x
+
⋯
+
n
a
n
x
n
−
1
f
(
x
)
(
f
+
g
)
′
(
x
)
=
f
′
(
x
)
+
g
′
(
x
)
D
:
F
[
x
]
→
F
[
x
]
D
(
f
(
x
)
)
=
f
′
(
x
)
D
char
F
=
0
D
char
F
=
p
(
f
g
)
′
(
x
)
=
f
′
(
x
)
g
(
x
)
+
f
(
x
)
g
′
(
x
)
f
(
x
)
∈
F
[
x
]
f
(
x
)
=
a
(
x
−
a
1
)
(
x
−
a
2
)
⋯
(
x
−
a
n
)
f
(
x
)
f
(
x
)
f
′
(
x
)
F
F
[
x
]
F
[
x
]
R
R
[
x
1
,
…
,
x
n
]
R
R
[
x
]
R
′
R
p
(
x
)
q
(
x
)
R
[
x
]
R
deg
(
p
(
x
)
+
q
(
x
)
)
≤
max
(
deg
p
(
x
)
,
deg
q
(
x
)
)
a
x
2
+
b
x
+
c
=
0
x
=
−
b
±
b
2
−
4
a
c
2
a
Δ
=
b
2
−
4
a
c
Δ
>
0
Δ
=
0
Δ
<
0
x
3
+
b
x
2
+
c
x
+
d
=
0
y
3
+
p
y
+
q
=
0
x
=
y
−
b
/
3
ω
=
−
1
+
i
3
2
ω
2
=
−
1
−
i
3
2
ω
3
=
1
y
=
z
−
p
3
z
y
y
3
+
p
y
+
q
=
0
A
B
z
3
−
p
3
/
27
A
B
3
=
−
p
/
3
z
A
3
,
ω
A
3
,
ω
2
A
3
,
B
3
,
ω
B
3
,
ω
2
B
3
y
ω
i
−
q
2
+
p
3
27
+
q
2
4
3
+
ω
2
i
−
q
2
−
p
3
27
+
q
2
4
3
i
=
0
,
1
,
2
Δ
=
p
3
27
+
q
2
4
y
3
+
p
y
+
q
=
0
Δ
=
0
Δ
>
0
Δ
<
0
x
3
−
4
x
2
+
11
x
+
30
=
0
x
3
−
3
x
+
5
=
0
x
3
−
3
x
+
2
=
0
x
3
+
x
+
3
=
0
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
=
0
y
4
+
p
y
2
+
q
y
+
r
=
0
x
=
y
−
a
/
4
(
y
2
+
1
2
z
)
2
=
(
z
−
p
)
y
2
−
q
y
+
(
1
4
z
2
−
r
)
(
m
y
+
k
)
2
q
2
−
4
(
z
−
p
)
(
1
4
z
2
−
r
)
=
0
z
3
−
p
z
2
−
4
r
z
+
(
4
p
r
−
q
2
)
=
0
(
y
2
+
1
2
z
)
2
=
(
m
y
+
k
)
2
x
4
−
x
2
−
3
x
+
2
=
0
x
4
+
x
3
−
7
x
2
−
x
+
6
=
0
x
4
−
2
x
2
+
4
x
−
3
=
0
x
4
−
4
x
3
+
3
x
2
−
5
x
+
2
=
0
a
2
+
4
a
+
2
=
0
a
2
=
−
4
a
−
3
=
a
+
2
a
2
a
+
2
Z
p
p
n
Z
p
F
F
[
x
]
7
5
=
16
807
5
Z
7
x
+
⟨
x
5
+
x
+
4
⟩
x
3
−
3
x
+
4
Z
5
Z
p
n
Z
p
Z
5
p
=
x
4
+
4
x
2
+
4
x
+
2
x
Z
5
Z
5
729
p
=
x
3
+
2
x
2
+
2
x
+
4
q
=
x
4
+
2
x
2
r
(
x
)
s
(
x
)
r
(
x
)
p
(
x
)
+
s
(
x
)
q
(
x
)
Z
[
x
]
Q
Z
Q
D
F
D
p
/
q
∈
Q
p
q
1
/
2
=
2
/
4
=
3
/
6
a
b
=
c
d
a
d
=
b
c
Q
Z
×
Z
p
/
q
(
p
,
q
)
(
3
,
7
)
3
/
7
Z
×
Z
5
/
0
(
5
,
0
)
(
3
,
6
)
(
2
,
4
)
1
/
2
(
a
,
b
)
(
c
,
d
)
a
d
=
b
c
D
S
=
{
(
a
,
b
)
:
a
,
b
∈
D
and
b
0
}
S
(
a
,
b
)
∼
(
c
,
d
)
a
d
=
b
c
∼
S
D
a
b
=
b
a
∼
D
(
a
,
b
)
∼
(
c
,
d
)
a
d
=
b
c
c
b
=
d
a
(
c
,
d
)
∼
(
a
,
b
)
(
a
,
b
)
∼
(
c
,
d
)
(
c
,
d
)
∼
(
e
,
f
)
a
d
=
b
c
c
f
=
d
e
a
d
=
b
c
f
a
f
d
=
a
d
f
=
b
c
f
=
b
d
e
=
b
e
d
D
a
f
=
b
e
(
a
,
b
)
∼
(
e
,
f
)
S
F
D
F
D
Q
a
b
+
c
d
=
a
d
+
b
c
b
d
;
a
b
⋅
c
d
=
a
c
b
d
F
D
(
a
,
b
)
∈
S
[
a
,
b
]
F
D
[
a
,
b
]
+
[
c
,
d
]
=
[
a
d
+
b
c
,
b
d
]
[
a
,
b
]
⋅
[
c
,
d
]
=
[
a
c
,
b
d
]
F
D
[
a
1
,
b
1
]
=
[
a
2
,
b
2
]
[
c
1
,
d
1
]
=
[
c
2
,
d
2
]
[
a
1
d
1
+
b
1
c
1
,
b
1
d
1
]
=
[
a
2
d
2
+
b
2
c
2
,
b
2
d
2
]
(
a
1
d
1
+
b
1
c
1
)
(
b
2
d
2
)
=
(
b
1
d
1
)
(
a
2
d
2
+
b
2
c
2
)
[
a
1
,
b
1
]
=
[
a
2
,
b
2
]
[
c
1
,
d
1
]
=
[
c
2
,
d
2
]
a
1
b
2
=
b
1
a
2
c
1
d
2
=
d
1
c
2
(
a
1
d
1
+
b
1
c
1
)
(
b
2
d
2
)
=
a
1
d
1
b
2
d
2
+
b
1
c
1
b
2
d
2
=
a
1
b
2
d
1
d
2
+
b
1
b
2
c
1
d
2
=
b
1
a
2
d
1
d
2
+
b
1
b
2
d
1
c
2
=
(
b
1
d
1
)
(
a
2
d
2
+
b
2
c
2
)
S
F
D
∼
[
a
,
b
]
+
[
c
,
d
]
=
[
a
d
+
b
c
,
b
d
]
[
a
,
b
]
⋅
[
c
,
d
]
=
[
a
c
,
b
d
]
[
0
,
1
]
[
1
,
1
]
[
0
,
1
]
[
a
,
b
]
+
[
0
,
1
]
=
[
a
1
+
b
0
,
b
1
]
=
[
a
,
b
]
[
1
,
1
]
[
a
,
b
]
∈
F
D
a
0
[
b
,
a
]
F
D
[
a
,
b
]
⋅
[
b
,
a
]
=
[
1
,
1
]
[
b
,
a
]
[
a
,
b
]
[
−
a
,
b
]
[
a
,
b
]
F
D
F
D
F
D
[
a
,
b
]
[
e
,
f
]
+
[
c
,
d
]
[
e
,
f
]
=
[
a
e
,
b
f
]
+
[
c
e
,
d
f
]
=
[
a
e
d
f
+
b
f
c
e
,
b
d
f
2
]
=
[
a
e
d
+
b
c
e
,
b
d
f
]
=
[
a
d
e
+
b
c
e
,
b
d
f
]
=
(
[
a
,
b
]
+
[
c
,
d
]
)
[
e
,
f
]
F
D
D
D
D
F
D
F
D
D
F
D
E
D
ψ
:
F
D
→
E
E
ψ
(
a
)
=
a
a
∈
D
a
F
D
D
F
D
ϕ
:
D
→
F
D
ϕ
(
a
)
=
[
a
,
1
]
a
b
D
ϕ
(
a
+
b
)
=
[
a
+
b
,
1
]
=
[
a
,
1
]
+
[
b
,
1
]
=
ϕ
(
a
)
+
ϕ
(
b
)
ϕ
(
a
b
)
=
[
a
b
,
1
]
=
[
a
,
1
]
[
b
,
1
]
=
ϕ
(
a
)
ϕ
(
b
)
;
ϕ
ϕ
ϕ
(
a
)
=
ϕ
(
b
)
[
a
,
1
]
=
[
b
,
1
]
a
=
a
1
=
1
b
=
b
F
D
D
ϕ
(
a
)
[
ϕ
(
b
)
]
−
1
=
[
a
,
1
]
[
b
,
1
]
−
1
=
[
a
,
1
]
⋅
[
1
,
b
]
=
[
a
,
b
]
E
D
ψ
:
F
D
→
E
ψ
(
[
a
,
b
]
)
=
a
b
−
1
ψ
[
a
1
,
b
1
]
=
[
a
2
,
b
2
]
a
1
b
2
=
b
1
a
2
a
1
b
1
−
1
=
a
2
b
2
−
1
ψ
(
[
a
1
,
b
1
]
)
=
ψ
(
[
a
2
,
b
2
]
)
[
a
,
b
]
[
c
,
d
]
F
D
ψ
(
[
a
,
b
]
+
[
c
,
d
]
)
=
ψ
(
[
a
d
+
b
c
,
b
d
]
)
=
(
a
d
+
b
c
)
(
b
d
)
−
1
=
a
b
−
1
+
c
d
−
1
=
ψ
(
[
a
,
b
]
)
+
ψ
(
[
c
,
d
]
)
ψ
(
[
a
,
b
]
⋅
[
c
,
d
]
)
=
ψ
(
[
a
c
,
b
d
]
)
=
(
a
c
)
(
b
d
)
−
1
=
a
b
−
1
c
d
−
1
=
ψ
(
[
a
,
b
]
)
ψ
(
[
c
,
d
]
)
ψ
ψ
ψ
(
[
a
,
b
]
)
=
a
b
−
1
=
0
a
=
0
b
=
0
[
a
,
b
]
=
[
0
,
b
]
ψ
[
0
,
b
]
F
D
ψ
Q
Q
[
x
]
Q
[
x
]
p
(
x
)
/
q
(
x
)
p
(
x
)
q
(
x
)
q
(
x
)
Q
(
x
)
Q
F
F
Q
F
p
F
Z
p
F
F
[
x
]
R
a
b
R
a
b
a
∣
b
c
∈
R
b
=
a
c
R
a
b
R
u
R
a
=
u
b
D
p
∈
D
p
=
a
b
a
b
p
p
∣
a
b
p
∣
a
p
∣
b
R
Q
[
x
,
y
]
x
2
y
2
x
y
R
x
y
x
y
x
2
y
2
x
2
y
2
n
>
1
p
1
⋯
p
k
p
i
p
i
D
D
a
∈
D
a
0
a
a
D
a
=
p
1
⋯
p
r
=
q
1
⋯
q
s
p
i
q
i
r
=
s
π
∈
S
r
p
i
q
π
(
j
)
j
=
1
,
…
,
r
Z
[
3
i
]
=
{
a
+
b
3
i
}
z
=
a
+
b
3
i
ν
:
Z
[
3
i
]
→
N
∪
{
0
}
ν
(
z
)
=
|
z
|
2
=
a
2
+
3
b
2
ν
(
z
)
≥
0
z
=
0
ν
(
z
w
)
=
ν
(
z
)
ν
(
w
)
ν
(
z
)
=
1
z
Z
[
3
i
]
1
−
1
4
4
=
2
⋅
2
=
(
1
−
3
i
)
(
1
+
3
i
)
Z
[
3
i
]
2
2
=
z
w
z
,
w
Z
[
3
i
]
ν
(
z
)
=
ν
(
w
)
=
2
z
Z
[
3
i
]
ν
(
z
)
=
2
a
2
+
3
b
2
=
2
2
1
−
3
i
1
+
3
i
2
1
−
3
i
1
+
3
i
4
R
a
∈
R
⟨
a
⟩
=
{
r
a
:
r
∈
R
}
D
a
,
b
∈
D
a
∣
b
⟨
b
⟩
⊂
⟨
a
⟩
a
b
⟨
b
⟩
=
⟨
a
⟩
a
D
⟨
a
⟩
=
D
a
∣
b
b
=
a
x
x
∈
D
r
D
b
r
=
(
a
x
)
r
=
a
(
x
r
)
⟨
b
⟩
⊂
⟨
a
⟩
⟨
b
⟩
⊂
⟨
a
⟩
b
∈
⟨
a
⟩
b
=
a
x
x
∈
D
a
∣
b
a
b
u
a
=
u
b
b
∣
a
⟨
a
⟩
⊂
⟨
b
⟩
⟨
b
⟩
⊂
⟨
a
⟩
⟨
a
⟩
=
⟨
b
⟩
⟨
a
⟩
=
⟨
b
⟩
a
∣
b
b
∣
a
a
=
b
x
b
=
a
y
x
,
y
∈
D
a
=
b
x
=
a
y
x
D
x
y
=
1
x
y
a
b
a
∈
D
a
1
a
1
⟨
a
⟩
=
⟨
1
⟩
=
D
D
⟨
p
⟩
D
⟨
p
⟩
p
⟨
p
⟩
a
D
p
⟨
p
⟩
⊂
⟨
a
⟩
⟨
p
⟩
D
=
⟨
a
⟩
⟨
p
⟩
=
⟨
a
⟩
a
p
a
p
p
⟨
a
⟩
D
⟨
p
⟩
⊂
⟨
a
⟩
⊂
D
a
∣
p
p
a
a
p
D
=
⟨
a
⟩
⟨
p
⟩
=
⟨
a
⟩
⟨
p
⟩
D
p
p
p
p
∣
a
b
⟨
a
b
⟩
⊂
⟨
p
⟩
⟨
p
⟩
⟨
p
⟩
a
∈
⟨
p
⟩
b
∈
⟨
p
⟩
p
∣
a
p
∣
b
D
I
1
,
I
2
,
…
I
1
⊂
I
2
⊂
⋯
N
I
n
=
I
N
n
≥
N
I
=
⋃
i
=
1
∞
I
i
D
I
I
1
⊂
I
0
∈
I
a
,
b
∈
I
a
∈
I
i
b
∈
I
j
i
j
N
i
≤
j
a
b
I
j
a
−
b
I
j
r
∈
D
a
∈
I
a
∈
I
i
i
I
i
r
a
∈
I
i
I
I
D
D
a
¯
∈
D
I
a
¯
I
N
N
∈
N
I
N
=
I
=
⟨
a
¯
⟩
I
n
=
I
N
n
≥
N
D
a
D
a
a
=
a
1
b
1
a
1
b
1
⟨
a
⟩
⊂
⟨
a
1
⟩
⟨
a
⟩
⟨
a
1
⟩
a
a
1
b
1
a
1
=
a
2
b
2
a
2
b
2
⟨
a
1
⟩
⊂
⟨
a
2
⟩
⟨
a
⟩
⊂
⟨
a
1
⟩
⊂
⟨
a
2
⟩
⊂
⋯
N
⟨
a
n
⟩
=
⟨
a
N
⟩
n
≥
N
a
N
a
a
=
c
1
p
1
p
1
c
1
⟨
a
⟩
⊂
⟨
c
1
⟩
c
1
c
1
=
c
2
p
2
p
2
c
2
⟨
a
⟩
⊂
⟨
c
1
⟩
⊂
⟨
c
2
⟩
⊂
⋯
a
=
p
1
p
2
⋯
p
r
p
1
,
…
,
p
r
a
=
p
1
p
2
⋯
p
r
=
q
1
q
2
⋯
q
s
p
i
q
i
r
<
s
p
1
q
1
q
2
⋯
q
s
q
i
q
i
p
1
∣
q
1
q
1
=
u
1
p
1
u
1
D
a
=
p
1
p
2
⋯
p
r
=
u
1
p
1
q
2
⋯
q
s
p
2
⋯
p
r
=
u
1
q
2
⋯
q
s
q
i
p
2
=
q
2
,
p
3
=
q
3
,
…
,
p
r
=
q
r
u
1
u
2
⋯
u
r
q
r
+
1
⋯
q
s
=
1
q
r
+
1
⋯
q
s
q
r
+
1
,
…
,
q
s
r
=
s
a
F
F
[
x
]
Z
[
x
]
Z
[
x
]
I
=
{
5
f
(
x
)
+
x
g
(
x
)
:
f
(
x
)
,
g
(
x
)
∈
Z
[
x
]
}
I
Z
[
x
]
I
=
⟨
p
(
x
)
⟩
5
∈
I
5
=
f
(
x
)
p
(
x
)
p
(
x
)
=
p
x
∈
I
x
=
p
g
(
x
)
p
=
±
1
⟨
p
(
x
)
⟩
=
Z
[
x
]
3
I
3
=
5
f
(
x
)
+
x
g
(
x
)
f
(
x
)
g
(
x
)
Z
[
x
]
3
=
5
f
(
x
)
Z
F
[
x
]
F
D
ν
:
D
∖
{
0
}
→
N
a
a
b
D
ν
(
a
)
≤
ν
(
a
b
)
a
,
b
∈
D
b
0
q
,
r
∈
D
a
=
b
q
+
r
r
=
0
ν
(
r
)
<
ν
(
b
)
D
ν
Z
F
F
[
x
]
Z
[
i
]
=
{
a
+
b
i
:
a
,
b
∈
Z
}
a
+
b
i
|
a
+
b
i
|
=
a
2
+
b
2
a
2
+
b
2
ν
(
a
+
b
i
)
=
a
2
+
b
2
ν
(
a
+
b
i
)
=
a
2
+
b
2
Z
[
i
]
z
,
w
∈
Z
[
i
]
ν
(
z
w
)
=
|
z
w
|
2
=
|
z
|
2
|
w
|
2
=
ν
(
z
)
ν
(
w
)
ν
(
z
)
≥
1
z
∈
Z
[
i
]
ν
(
z
)
≤
ν
(
z
)
ν
(
w
)
z
=
a
+
b
i
w
=
c
+
d
i
Z
[
i
]
w
0
q
r
Z
[
i
]
z
=
q
w
+
r
r
=
0
ν
(
r
)
<
ν
(
w
)
z
w
Q
(
i
)
=
{
p
+
q
i
:
p
,
q
∈
Q
}
Z
[
i
]
z
w
−
1
=
(
a
+
b
i
)
c
−
d
i
c
2
+
d
2
=
a
c
+
b
d
c
2
+
d
2
+
b
c
−
a
d
c
2
+
d
2
i
=
(
m
1
+
n
1
c
2
+
d
2
)
+
(
m
2
+
n
2
c
2
+
d
2
)
i
=
(
m
1
+
m
2
i
)
+
(
n
1
c
2
+
d
2
+
n
2
c
2
+
d
2
i
)
=
(
m
1
+
m
2
i
)
+
(
s
+
t
i
)
Q
(
i
)
m
i
|
n
i
/
(
a
2
+
b
2
)
|
≤
1
/
2
9
8
=
1
+
1
8
15
8
=
2
−
1
8
s
t
z
w
−
1
=
(
m
1
+
m
2
i
)
+
(
s
+
t
i
)
s
2
+
t
2
≤
1
/
4
+
1
/
4
=
1
/
2
w
z
=
z
w
−
1
w
=
w
(
m
1
+
m
2
i
)
+
w
(
s
+
t
i
)
=
q
w
+
r
q
=
m
1
+
m
2
i
r
=
w
(
s
+
t
i
)
z
q
w
Z
[
i
]
r
Z
[
i
]
r
=
0
ν
(
r
)
<
ν
(
w
)
ν
(
r
)
=
ν
(
w
)
ν
(
s
+
t
i
)
≤
1
2
ν
(
w
)
<
ν
(
w
)
D
ν
D
I
D
b
∈
I
ν
(
b
)
a
∈
I
D
q
r
D
a
=
b
q
+
r
r
=
0
ν
(
r
)
<
ν
(
b
)
r
=
a
−
b
q
I
I
r
=
0
b
a
=
b
q
I
=
⟨
b
⟩
D
[
x
]
Z
[
x
]
Z
[
x
]
D
p
(
x
)
=
a
n
x
n
+
⋯
+
a
1
x
+
a
0
D
[
x
]
p
(
x
)
a
0
,
…
,
a
n
p
(
x
)
gcd
(
a
0
,
…
,
a
n
)
=
1
Z
[
x
]
p
(
x
)
=
5
x
4
−
3
x
3
+
x
−
4
1
q
(
x
)
=
4
x
2
−
6
x
+
8
q
(
x
)
2
D
f
(
x
)
g
(
x
)
D
[
x
]
f
(
x
)
g
(
x
)
f
(
x
)
=
∑
i
=
0
m
a
i
x
i
g
(
x
)
=
∑
i
=
0
n
b
i
x
i
p
f
(
x
)
g
(
x
)
r
p
∤
a
r
s
p
∤
b
s
x
r
+
s
f
(
x
)
g
(
x
)
c
r
+
s
=
a
0
b
r
+
s
+
a
1
b
r
+
s
−
1
+
⋯
+
a
r
+
s
−
1
b
1
+
a
r
+
s
b
0
p
a
0
,
…
,
a
r
−
1
b
0
,
…
,
b
s
−
1
p
c
r
+
s
a
r
b
s
p
∣
c
r
+
s
p
a
r
p
b
s
D
p
(
x
)
q
(
x
)
D
[
x
]
p
(
x
)
q
(
x
)
p
(
x
)
q
(
x
)
p
(
x
)
=
c
p
1
(
x
)
q
(
x
)
=
d
q
1
(
x
)
c
d
p
(
x
)
q
(
x
)
p
1
(
x
)
q
1
(
x
)
p
(
x
)
q
(
x
)
=
c
d
p
1
(
x
)
q
1
(
x
)
p
1
(
x
)
q
1
(
x
)
p
(
x
)
q
(
x
)
c
d
D
F
p
(
x
)
∈
D
[
x
]
p
(
x
)
=
f
(
x
)
g
(
x
)
f
(
x
)
g
(
x
)
F
[
x
]
p
(
x
)
=
f
1
(
x
)
g
1
(
x
)
f
1
(
x
)
g
1
(
x
)
D
[
x
]
deg
f
(
x
)
=
deg
f
1
(
x
)
deg
g
(
x
)
=
deg
g
1
(
x
)
a
b
D
a
f
(
x
)
,
b
g
(
x
)
D
[
x
]
a
1
,
b
1
∈
D
a
f
(
x
)
=
a
1
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1
(
x
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b
g
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=
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(
x
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1
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1
(
x
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D
[
x
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a
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x
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a
1
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1
(
x
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b
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1
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1
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1
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x
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a
b
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1
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1
c
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p
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x
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f
1
(
x
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g
1
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deg
f
(
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=
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1
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deg
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deg
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1
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p
(
x
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D
[
x
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x
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x
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p
(
x
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[
x
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(
x
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(
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(
x
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x
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(
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1
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1
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1
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D
[
x
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1
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D
D
[
x
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(
x
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[
x
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(
x
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(
x
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[
x
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D
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(
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x
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D
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(
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p
(
x
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D
[
x
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x
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3
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(
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Q
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b
2
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a
,
b
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[
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Z
[
2
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Z
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2
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Z
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2
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a
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b
2
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d
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d
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a
b
D
a
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′
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b
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a
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s
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ν
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b
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Z
[
5
i
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1
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a
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r
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1
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a
n
r
n
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1
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…
,
r
n
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R
R
D
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1
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k
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k
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b
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S
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S
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1
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R
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→
S
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1
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(
a
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1
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0
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ψ
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R
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1
R
p
Z
p
4
Z
[
3
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N
Z
Q
R
X
X
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X
P
X
X
(
a
,
a
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P
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X
(
a
,
b
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P
(
b
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a
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P
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b
(
a
,
b
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P
(
b
,
c
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P
(
a
,
c
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P
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b
(
a
,
b
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P
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a
b
A
⊂
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B
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b
a
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a
b
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X
X
X
X
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(
X
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X
=
{
a
,
b
,
c
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P
(
X
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{
a
,
b
,
c
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∅
{
a
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{
b
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{
c
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{
a
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c
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b
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(
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b
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a
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b
a
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a
a
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m
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n
n
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m
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m
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n
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p
m
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X
=
{
1
,
2
,
3
,
4
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6
,
8
,
12
,
24
}
24
X
24
Y
X
u
X
Y
a
⪯
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a
∈
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u
Y
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Y
l
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Y
l
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a
a
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l
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k
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k
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Y
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=
{
2
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3
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4
,
6
}
X
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12
24
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1
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Y
Y
Y
Y
u
1
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1
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1
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1
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1
=
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L
a
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b
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b
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b
a
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b
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P
(
X
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B
P
(
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A
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A
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(
A
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P
(
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(
X
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A
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P
(
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A
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L
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;
P
(
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(
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C
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(
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L
a
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b
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c
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L
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B
I
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(
X
)
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b
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a
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=
b
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a
a
,
b
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a
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(
b
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c
)
=
(
a
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b
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b
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c
)
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(
a
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b
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c
a
,
b
,
c
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B
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∧
(
b
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c
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=
(
a
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b
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a
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c
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b
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c
)
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(
a
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b
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(
a
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c
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a
,
b
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a
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a
a
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a
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B
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′
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′
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a
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b
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I
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b
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a
∧
I
=
a
B
B
B
a
∈
B
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a
=
a
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a
O
B
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B
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b
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a
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a
a
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(
a
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a
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a
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B
B
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b
a
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b
{
a
,
b
}
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B
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b
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a
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c
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b
=
a
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c
a
,
b
,
c
∈
B
b
=
c
a
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b
=
I
a
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b
=
O
b
=
a
′
(
a
′
)
′
=
a
a
∈
B
I
′
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O
O
′
=
I
(
a
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b
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′
=
a
′
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b
′
(
a
∧
b
)
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=
a
′
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b
′
a
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b
=
a
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c
a
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b
=
a
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c
b
=
b
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(
b
∧
a
)
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b
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a
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b
)
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b
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a
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c
)
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(
b
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a
)
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(
b
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c
)
=
(
a
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b
)
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(
b
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c
)
=
(
a
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c
)
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(
b
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c
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(
c
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c
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b
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(
a
∧
c
)
=
c
∨
(
c
∧
a
)
=
c
B
C
ϕ
:
B
→
C
ϕ
(
a
∨
b
)
=
ϕ
(
a
)
∨
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(
b
)
ϕ
(
a
∧
b
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(
a
)
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ϕ
(
b
)
a
b
B
X
B
a
∈
B
B
a
O
a
∧
b
=
a
b
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b
O
a
B
b
∈
B
b
O
a
O
⪯
b
⪯
a
B
b
B
b
O
a
B
a
⪯
b
b
a
=
b
b
1
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b
b
1
⪯
b
b
b
1
b
2
O
b
1
b
2
⪯
b
1
b
2
a
=
b
2
O
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⋯
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b
3
⪯
b
2
⪯
b
1
⪯
b
B
k
b
k
a
=
b
k
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b
B
a
b
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a
b
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a
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b
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a
a
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=
a
a
⪯
b
a
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a
b
a
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b
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a
,
b
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b
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c
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a
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b
a
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b
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a
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a
n
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i
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b
b
=
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1
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a
n
a
,
a
1
,
…
,
a
n
B
a
⪯
b
a
i
⪯
b
b
=
a
∨
a
1
∨
⋯
∨
a
n
a
=
a
i
i
=
1
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…
,
n
b
1
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a
1
∨
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a
n
a
i
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b
i
b
1
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b
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1
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1
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b
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i
a
i
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1
b
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a
1
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n
a
b
a
=
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∧
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=
a
∧
(
a
1
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⋯
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a
n
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=
(
a
∧
a
1
)
∨
⋯
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(
a
∧
a
n
)
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a
a
∧
a
i
a
i
a
=
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i
i
B
X
B
P
(
X
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B
P
(
X
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X
B
a
∈
B
a
a
=
a
1
∨
⋯
∨
a
n
a
1
,
…
,
a
n
∈
X
ϕ
:
B
→
P
(
X
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ϕ
(
a
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=
ϕ
(
a
1
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⋯
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a
n
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=
{
a
1
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…
,
a
n
}
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=
a
1
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a
n
b
=
b
1
∨
⋯
∨
b
m
B
a
i
b
i
ϕ
(
a
)
=
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(
b
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{
a
1
,
…
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a
n
}
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{
b
1
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…
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b
m
}
a
=
b
ϕ
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b
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(
a
∨
b
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(
a
1
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a
n
∨
b
1
∨
⋯
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b
m
)
=
{
a
1
,
…
,
a
n
,
b
1
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…
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m
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{
a
1
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…
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a
n
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{
b
1
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…
,
b
m
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=
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(
a
1
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⋯
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a
n
)
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ϕ
(
b
1
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⋯
∨
b
m
)
=
ϕ
(
a
)
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ϕ
(
b
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ϕ
(
a
∧
b
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=
ϕ
(
a
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ϕ
(
b
)
2
n
n
a
a
b
A
B
a
b
a
∧
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a
b
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B
a
b
a
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a
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b
a
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b
b
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a
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a
∧
(
b
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(
a
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b
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(
a
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c
)
a
a
′
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a
I
O
a
∧
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a
′
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I
a
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(
b
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c
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(
a
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b
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(
a
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c
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′
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(
a
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b
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(
a
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b
′
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(
a
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b
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(
a
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b
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(
a
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b
′
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(
a
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b
)
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(
a
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b
)
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(
a
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b
)
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(
a
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b
′
)
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(
a
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b
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(
a
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b
′
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a
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(
b
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b
′
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=
a
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=
a
a
(
a
∨
b
)
∧
(
a
∨
b
′
)
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(
a
∨
b
)
X
=
{
a
,
b
,
c
,
d
}
⊂
30
Z
12
B
210
B
a
⪯
b
a
∣
b
B
X
B
P
(
X
)
B
Z
a
⪯
b
a
∣
b
(
a
∨
b
∨
a
′
)
∧
a
(
a
∨
b
)
′
∧
(
a
∨
b
)
a
∨
(
a
∧
b
)
(
c
∨
a
∨
b
)
∧
c
′
∧
(
a
∨
b
)
′
(
a
∨
b
∨
a
′
)
∧
a
a
∨
(
a
∧
b
)
a
b
c
X
n
|
P
(
X
)
|
=
2
n
2
n
n
∈
N
a
′
∧
[
(
a
∧
b
′
)
∨
b
]
=
a
∧
(
a
∨
b
)
a
⪯
b
a
∣
b
L
∨
∧
L
a
⪯
b
a
∨
b
=
b
a
b
a
∧
b
G
X
G
H
K
G
H
K
H
∪
K
R
X
R
X
⊂
I
J
X
I
∩
J
I
J
I
+
J
R
I
,
J
R
I
+
J
=
{
r
+
s
:
r
∈
I
and
s
∈
J
}
R
I
J
r
1
,
r
2
∈
I
s
1
,
s
2
∈
J
(
r
1
+
s
1
)
+
(
r
2
+
s
2
)
=
(
r
1
+
r
2
)
+
(
s
1
+
s
2
)
I
+
J
a
∈
R
a
(
r
1
+
s
1
)
=
a
r
1
+
a
s
1
∈
I
+
J
I
+
J
R
B
a
∨
I
=
I
a
∧
O
=
O
a
∈
B
a
∨
b
=
I
a
∧
b
=
O
b
=
a
′
(
a
′
)
′
=
a
a
∈
B
I
′
=
O
O
′
=
I
(
a
∨
b
)
′
=
a
′
∧
b
′
(
a
∧
b
)
′
=
a
′
∨
b
′
B
+
⋅
B
a
+
b
=
(
a
∧
b
′
)
∨
(
a
′
∧
b
)
a
⋅
b
=
a
∧
b
B
a
2
=
a
a
∈
B
X
a
b
X
a
⪯
b
b
⪯
a
X
a
∣
b
N
N
Z
Q
R
≤
X
Y
ϕ
:
X
→
Y
a
⪯
b
ϕ
(
a
)
⪯
ϕ
(
b
)
L
M
ψ
:
L
→
M
ψ
(
a
∨
b
)
=
ψ
(
a
)
∨
ψ
(
b
)
ψ
(
a
∧
b
)
=
ψ
(
a
)
∧
ψ
(
b
)
B
a
=
b
(
a
∧
b
′
)
∨
(
a
′
∧
b
)
=
O
a
,
b
∈
B
(
⇒
)
a
=
b
⇒
(
a
∧
b
′
)
∨
(
a
′
∧
b
)
=
(
a
∧
a
′
)
∨
(
a
′
∧
a
)
=
O
∨
O
=
O
(
⇐
)
(
a
∧
b
′
)
∨
(
a
′
∧
b
)
=
O
⇒
a
∨
b
=
(
a
∨
a
)
∨
b
=
a
∨
(
a
∨
b
)
=
a
∨
[
I
∧
(
a
∨
b
)
]
=
a
∨
[
(
a
∨
a
′
)
∧
(
a
∨
b
)
]
=
[
a
∨
(
a
∧
b
′
)
]
∨
[
a
∨
(
a
′
∧
b
)
]
=
a
∨
[
(
a
∧
b
′
)
∨
(
a
′
∧
b
)
]
=
a
∨
0
=
a
a
∨
b
=
b
B
a
=
O
(
a
∧
b
′
)
∨
(
a
′
∧
b
)
=
b
b
∈
B
L
M
L
×
M
(
a
,
b
)
⪯
(
c
,
d
)
a
⪯
c
b
⪯
d
L
×
M
n
f
:
{
O
,
I
}
n
→
{
0
,
I
}
x
1
,
…
,
x
n
O
I
∨
∧
′
x
y
x
′
x
∨
y
x
∧
y
0
0
1
0
0
0
1
1
1
0
1
0
0
1
0
1
1
0
1
1
X
n
n
n
C
1
C
2
C
2
C
1
C
1
=
[
2
,
1
,
2
]
⪰
[
3
,
2
]
=
C
2
n
5
1
72
=
2
3
⋅
3
2
n
16
4
16
5
x
y
z
n
V
F
α
⋅
v
α
v
α
∈
F
v
∈
V
α
(
β
v
)
=
(
α
β
)
v
(
α
+
β
)
v
=
α
v
+
β
v
α
(
u
+
v
)
=
α
u
+
α
v
1
v
=
v
α
,
β
∈
F
u
,
v
∈
V
V
F
0
n
R
n
R
u
=
(
u
1
,
…
,
u
n
)
v
=
(
v
1
,
…
,
v
n
)
R
n
α
R
u
+
v
=
(
u
1
,
…
,
u
n
)
+
(
v
1
,
…
,
v
n
)
=
(
u
1
+
v
1
,
…
,
u
n
+
v
n
)
α
u
=
α
(
u
1
,
…
,
u
n
)
=
(
α
u
1
,
…
,
α
u
n
)
F
F
[
x
]
F
F
[
x
]
α
∈
F
p
(
x
)
∈
F
[
x
]
α
p
(
x
)
[
a
,
b
]
R
f
(
x
)
g
(
x
)
[
a
,
b
]
(
f
+
g
)
(
x
)
f
(
x
)
+
g
(
x
)
(
α
f
)
(
x
)
=
α
f
(
x
)
α
∈
R
f
(
x
)
=
sin
x
g
(
x
)
=
x
2
(
2
f
+
5
g
)
(
x
)
=
2
sin
x
+
5
x
2
V
=
Q
(
2
)
=
{
a
+
b
2
:
a
,
b
∈
Q
}
V
Q
u
=
a
+
b
2
v
=
c
+
d
2
u
+
v
=
(
a
+
c
)
+
(
b
+
d
)
2
V
α
∈
Q
α
v
V
V
V
F
0
v
=
0
v
∈
V
α
0
=
0
α
∈
F
α
v
=
0
α
=
0
v
=
0
(
−
1
)
v
=
−
v
v
∈
V
−
(
α
v
)
=
(
−
α
)
v
=
α
(
−
v
)
α
∈
F
v
∈
V
0
v
=
(
0
+
0
)
v
=
0
v
+
0
v
;
0
+
0
v
=
0
v
+
0
v
V
0
=
0
v
α
=
0
α
0
α
v
=
0
1
/
α
v
=
0
v
+
(
−
1
)
v
=
1
v
+
(
−
1
)
v
=
(
1
−
1
)
v
=
0
v
=
0
−
v
=
(
−
1
)
v
V
F
W
V
W
V
u
,
v
∈
W
α
∈
F
u
+
v
α
v
W
W
R
3
W
=
{
(
x
1
,
2
x
1
+
x
2
,
x
1
−
x
2
)
:
x
1
,
x
2
∈
R
}
W
R
3
α
(
x
1
,
2
x
1
+
x
2
,
x
1
−
x
2
)
=
(
α
x
1
,
α
(
2
x
1
+
x
2
)
,
α
(
x
1
−
x
2
)
)
=
(
α
x
1
,
2
(
α
x
1
)
+
α
x
2
,
α
x
1
−
α
x
2
)
W
W
u
=
(
x
1
,
2
x
1
+
x
2
,
x
1
−
x
2
)
v
=
(
y
1
,
2
y
1
+
y
2
,
y
1
−
y
2
)
W
u
+
v
=
(
x
1
+
y
1
,
2
(
x
1
+
y
1
)
+
(
x
2
+
y
2
)
,
(
x
1
+
y
1
)
−
(
x
2
+
y
2
)
)
W
F
[
x
]
p
(
x
)
q
(
x
)
p
(
x
)
+
q
(
x
)
α
p
(
x
)
∈
W
α
∈
F
p
(
x
)
∈
W
V
F
v
1
,
v
2
,
…
,
v
n
V
α
1
,
α
2
,
…
,
α
n
F
w
V
w
=
∑
i
=
1
n
α
i
v
i
=
α
1
v
1
+
α
2
v
2
+
⋯
+
α
n
v
n
v
1
,
v
2
,
…
,
v
n
v
1
,
v
2
,
…
,
v
n
v
1
,
v
2
,
…
,
v
n
W
v
1
,
v
2
,
…
,
v
n
W
v
1
,
v
2
,
…
,
v
n
S
=
{
v
1
,
v
2
,
…
,
v
n
}
V
S
V
u
v
S
v
i
u
=
α
1
v
1
+
α
2
v
2
+
⋯
+
α
n
v
n
v
=
β
1
v
1
+
β
2
v
2
+
⋯
+
β
n
v
n
u
+
v
=
(
α
1
+
β
1
)
v
1
+
(
α
2
+
β
2
)
v
2
+
⋯
+
(
α
n
+
β
n
)
v
n
v
i
α
∈
F
α
u
=
(
α
α
1
)
v
1
+
(
α
α
2
)
v
2
+
⋯
+
(
α
α
n
)
v
n
S
S
=
{
v
1
,
v
2
,
…
,
v
n
}
V
α
1
,
α
2
…
α
n
∈
F
α
i
α
1
v
1
+
α
2
v
2
+
⋯
+
α
n
v
n
=
0
S
S
S
α
1
v
1
+
α
2
v
2
+
⋯
+
α
n
v
n
=
0
α
1
=
α
2
=
⋯
=
α
n
=
0
{
α
1
,
α
2
…
α
n
}
{
v
1
,
v
2
,
…
,
v
n
}
v
=
α
1
v
1
+
α
2
v
2
+
⋯
+
α
n
v
n
=
β
1
v
1
+
β
2
v
2
+
⋯
+
β
n
v
n
α
1
=
β
1
,
α
2
=
β
2
,
…
,
α
n
=
β
n
v
=
α
1
v
1
+
α
2
v
2
+
⋯
+
α
n
v
n
=
β
1
v
1
+
β
2
v
2
+
⋯
+
β
n
v
n
(
α
1
−
β
1
)
v
1
+
(
α
2
−
β
2
)
v
2
+
⋯
+
(
α
n
−
β
n
)
v
n
=
0
v
1
,
…
,
v
n
α
i
−
β
i
=
0
i
=
1
,
…
,
n
{
v
1
,
v
2
,
…
,
v
n
}
V
v
i
{
v
1
,
v
2
,
…
,
v
n
}
α
1
,
…
,
α
n
α
1
v
1
+
α
2
v
2
+
⋯
+
α
n
v
n
=
0
α
i
α
k
0
v
k
=
−
α
1
α
k
v
1
−
⋯
−
α
k
−
1
α
k
v
k
−
1
−
α
k
+
1
α
k
v
k
+
1
−
⋯
−
α
n
α
k
v
n
v
k
=
β
1
v
1
+
⋯
+
β
k
−
1
v
k
−
1
+
β
k
+
1
v
k
+
1
+
⋯
+
β
n
v
n
β
1
v
1
+
⋯
+
β
k
−
1
v
k
−
1
−
v
k
+
β
k
+
1
v
k
+
1
+
⋯
+
β
n
v
n
=
0
V
n
m
>
n
m
V
{
e
1
,
e
2
,
…
,
e
n
}
V
V
{
e
1
,
e
2
,
…
,
e
n
}
V
e
1
=
(
1
,
0
,
0
)
e
2
=
(
0
,
1
,
0
)
e
3
=
(
0
,
0
,
1
)
R
3
R
3
(
x
1
,
x
2
,
x
3
)
R
3
x
1
e
1
+
x
2
e
2
+
x
3
e
3
e
1
,
e
2
,
e
3
e
1
,
e
2
,
e
3
R
3
{
(
3
,
2
,
1
)
,
(
3
,
2
,
0
)
,
(
1
,
1
,
1
)
}
R
3
Q
(
2
)
=
{
a
+
b
2
:
a
,
b
∈
Q
}
{
1
,
2
}
{
1
+
2
,
1
−
2
}
Q
(
2
)
R
3
Q
(
2
)
{
e
1
,
e
2
,
…
,
e
m
}
{
f
1
,
f
2
,
…
,
f
n
}
V
m
=
n
{
e
1
,
e
2
,
…
,
e
m
}
n
≤
m
{
f
1
,
f
2
,
…
,
f
n
}
m
≤
n
m
=
n
{
e
1
,
e
2
,
…
,
e
n
}
V
V
n
dim
V
=
n
V
V
n
S
=
{
v
1
,
…
,
v
n
}
V
S
V
S
=
{
v
1
,
…
,
v
n
}
V
S
V
S
=
{
v
1
,
…
,
v
k
}
V
k
<
n
v
k
+
1
,
…
,
v
n
{
v
1
,
…
,
v
k
,
v
k
+
1
,
…
,
v
n
}
V
V
=
Q
(
11
)
=
{
a
+
b
11
∣
a
,
b
∈
Q
}
S
=
{
u
}
u
=
3
+
2
7
11
∈
V
F
F
[
x
]
F
F
[
x
]
α
p
(
x
)
α
∈
F
Q
(
2
)
Q
(
2
,
3
)
a
+
b
2
+
c
3
+
d
6
a
,
b
,
c
,
d
Q
Q
(
2
,
3
)
4
Q
Q
(
2
,
3
)
Q
(
2
,
3
)
{
1
,
2
,
3
,
6
}
Q
2
R
P
n
n
F
[
x
]
P
n
P
n
{
1
,
x
,
x
2
,
…
,
x
n
−
1
}
P
n
F
n
F
F
n
u
=
(
u
1
,
…
,
u
n
)
v
=
(
v
1
,
…
,
v
n
)
F
n
α
F
u
+
v
=
(
u
1
,
…
,
u
n
)
+
(
v
1
,
…
,
v
n
)
=
(
u
1
+
v
1
,
…
,
u
n
+
v
n
)
α
u
=
α
(
u
1
,
…
,
u
n
)
=
(
α
u
1
,
…
,
α
u
n
)
F
n
n
R
3
{
(
x
1
,
x
2
,
x
3
)
:
3
x
1
−
2
x
2
+
x
3
=
0
}
{
(
x
1
,
x
2
,
x
3
)
:
3
x
1
+
4
x
3
=
0
,
2
x
1
−
x
2
+
x
3
=
0
}
{
(
x
1
,
x
2
,
x
3
)
:
x
1
−
2
x
2
+
2
x
3
=
2
}
{
(
x
1
,
x
2
,
x
3
)
:
3
x
1
−
2
x
2
2
=
0
}
2
{
(
1
,
0
,
−
3
)
,
(
0
,
1
,
2
)
}
(
x
,
y
,
z
)
∈
R
3
A
x
+
B
y
+
C
z
=
0
D
x
+
E
y
+
C
z
=
0
R
3
W
[
0
,
1
]
f
(
0
)
=
0
W
C
[
0
,
1
]
V
F
−
(
α
v
)
=
(
−
α
)
v
=
α
(
−
v
)
α
∈
F
v
∈
V
0
=
α
0
=
α
(
−
v
+
v
)
=
α
(
−
v
)
+
α
v
−
α
v
=
α
(
−
v
)
V
n
S
=
{
v
1
,
…
,
v
n
}
V
S
V
S
=
{
v
1
,
…
,
v
n
}
V
S
V
S
=
{
v
1
,
…
,
v
k
}
V
k
<
n
v
k
+
1
,
…
,
v
n
{
v
1
,
…
,
v
k
,
v
k
+
1
,
…
,
v
n
}
V
0
v
0
=
0
,
v
1
,
…
,
v
n
∈
V
α
0
0
,
α
1
,
…
,
α
n
∈
F
α
0
v
0
+
⋯
+
α
n
v
n
=
0
V
{
0
}
V
V
n
m
V
m
>
n
V
W
F
m
n
T
:
V
→
W
T
(
u
+
v
)
=
T
(
u
)
+
T
(
v
)
T
(
α
v
)
=
α
T
(
v
)
α
∈
F
u
,
v
∈
V
T
V
W
T
ker
(
T
)
=
{
v
∈
V
:
T
(
v
)
=
0
}
V
T
T
T
R
(
V
)
=
{
w
∈
W
:
T
(
v
)
=
w
for some
v
∈
V
}
W
T
:
V
→
W
ker
(
T
)
=
{
0
}
{
v
1
,
…
,
v
k
}
T
{
v
1
,
…
,
v
k
,
v
k
+
1
,
…
,
v
m
}
V
{
T
(
v
k
+
1
)
,
…
,
T
(
v
m
)
}
T
T
m
−
k
dim
V
=
dim
W
T
:
V
→
W
u
,
v
∈
ker
(
T
)
α
∈
F
T
(
u
+
v
)
=
T
(
u
)
+
T
(
v
)
=
0
T
(
α
v
)
=
α
T
(
v
)
=
α
0
=
0
u
+
v
,
α
v
∈
ker
(
T
)
ker
(
T
)
V
T
(
u
)
=
T
(
v
)
T
(
u
−
v
)
=
T
(
u
)
−
T
(
v
)
=
0
u
−
v
=
0
u
=
v
V
W
n
F
T
:
V
→
W
{
v
1
,
…
,
v
n
}
V
{
T
(
v
1
)
,
…
,
T
(
v
n
)
}
W
F
n
F
n
U
V
W
U
V
U
+
V
u
+
v
u
∈
U
v
∈
V
U
+
V
U
∩
V
W
U
+
V
=
W
U
∩
V
=
0
W
W
=
U
⊕
V
U
V
w
∈
W
w
=
u
+
v
u
∈
U
v
∈
V
U
k
W
n
V
n
−
k
W
=
U
⊕
V
V
U
V
W
dim
(
U
+
V
)
=
dim
U
+
dim
V
−
dim
(
U
∩
V
)
u
,
u
′
∈
U
v
,
v
′
∈
V
(
u
+
v
)
+
(
u
′
+
v
′
)
=
(
u
+
u
′
)
+
(
v
+
v
′
)
∈
U
+
V
α
(
u
+
v
)
=
α
u
+
α
v
∈
U
+
V
V
W
F
V
W
Hom
(
V
,
W
)
F
U
V
(
S
+
T
)
(
v
)
=
S
(
v
)
+
T
(
v
)
(
α
S
)
(
v
)
=
α
S
(
v
)
S
,
T
∈
Hom
(
V
,
W
)
α
∈
F
v
∈
V
V
F
V
V
∗
=
Hom
(
V
,
F
)
V
V
v
1
,
…
,
v
n
V
v
=
α
1
v
1
+
⋯
+
α
n
v
n
V
ϕ
i
:
V
→
F
ϕ
i
(
v
)
=
α
i
ϕ
i
V
∗
v
1
,
…
,
v
n
{
(
3
,
1
)
,
(
2
,
−
2
)
}
R
2
(
R
2
)
∗
V
n
F
V
∗
∗
V
∗
v
∈
V
λ
v
V
∗
∗
v
↦
λ
v
V
V
∗
∗
P
n
n
F
F
n
7
6
=
117
649
Z
7
{
1
,
a
,
a
2
,
a
3
,
a
4
,
a
5
}
(
Z
7
)
6
(
Z
7
)
6
U
W
V
U
+
W
U
+
W
=
{
u
+
w
∣
u
∈
U
,
w
∈
W
}
U
W
U
W
V
5
6
2
U
W
U
∩
W
U
+
W
8
Q
[
2
4
]
4
c
=
2
4
F
p
n
m
×
m
m
p
n
2
×
2
3
×
3
2
,
3
,
4
,
5
p
n
p
n
F
m
F
5
3
Z
5
a
F
M
3
×
3
Z
5
x
∈
F
{
1
,
a
,
a
2
}
M
M
F
{
1
,
a
,
a
2
}
R
R
F
F
M
a
a
↦
a
5
M
F
Q
R
F
p
(
x
)
∈
F
[
x
]
E
F
p
(
x
)
E
[
x
]
p
(
x
)
=
x
4
−
5
x
2
+
6
Q
[
x
]
p
(
x
)
(
x
2
−
2
)
(
x
2
−
3
)
Q
[
x
]
p
(
x
)
p
(
x
)
=
(
x
−
2
)
(
x
+
2
)
(
x
−
3
)
(
x
+
3
)
p
(
x
)
Q
(
2
)
=
{
a
+
b
2
:
a
,
b
∈
Q
}
F
E
F
F
E
F
F
⊂
E
F
=
Q
(
2
)
=
{
a
+
b
2
:
a
,
b
∈
Q
}
E
=
Q
(
2
+
3
)
Q
2
+
3
E
F
E
F
2
E
2
+
3
E
1
/
(
2
+
3
)
=
3
−
2
E
2
+
3
3
−
2
2
3
E
p
(
x
)
=
x
2
+
x
+
1
∈
Z
2
[
x
]
p
(
x
)
Z
2
Z
2
α
p
(
α
)
=
0
⟨
p
(
x
)
⟩
p
(
x
)
Z
2
[
x
]
/
⟨
p
(
x
)
⟩
f
(
x
)
+
⟨
p
(
x
)
⟩
Z
2
[
x
]
/
⟨
p
(
x
)
⟩
f
(
x
)
=
(
x
2
+
x
+
1
)
q
(
x
)
+
r
(
x
)
r
(
x
)
x
2
+
x
+
1
f
(
x
)
+
⟨
x
2
+
x
+
1
⟩
=
r
(
x
)
+
⟨
x
2
+
x
+
1
⟩
r
(
x
)
0
1
x
1
+
x
E
=
Z
2
[
x
]
/
⟨
x
2
+
x
+
1
⟩
Z
2
α
p
(
x
)
Z
2
(
α
)
0
+
0
α
=
0
1
+
0
α
=
1
0
+
1
α
=
α
1
+
1
α
=
1
+
α
α
2
+
α
+
1
=
0
(
1
+
α
)
2
(
1
+
α
)
(
1
+
α
)
=
1
+
α
+
α
+
(
α
)
2
=
α
E
Z
2
(
α
)
+
0
1
α
1
+
α
0
0
1
α
1
+
α
1
1
0
1
+
α
α
α
α
1
+
α
0
1
1
+
α
1
+
α
α
1
0
Z
2
(
α
)
⋅
0
1
α
1
+
α
0
0
0
0
0
1
0
1
α
1
+
α
α
0
α
1
+
α
1
1
+
α
0
1
+
α
1
α
F
p
(
x
)
F
[
x
]
E
F
α
∈
E
p
(
α
)
=
0
p
(
x
)
E
F
α
p
(
α
)
=
0
⟨
p
(
x
)
⟩
p
(
x
)
F
[
x
]
F
[
x
]
/
⟨
p
(
x
)
⟩
E
=
F
[
x
]
/
⟨
p
(
x
)
⟩
E
F
ψ
:
F
→
F
[
x
]
/
⟨
p
(
x
)
⟩
ψ
(
a
)
=
a
+
⟨
p
(
x
)
⟩
a
∈
F
ψ
ψ
(
a
)
+
ψ
(
b
)
=
(
a
+
⟨
p
(
x
)
⟩
)
+
(
b
+
⟨
p
(
x
)
⟩
)
=
(
a
+
b
)
+
⟨
p
(
x
)
⟩
=
ψ
(
a
+
b
)
ψ
(
a
)
ψ
(
b
)
=
(
a
+
⟨
p
(
x
)
⟩
)
(
b
+
⟨
p
(
x
)
⟩
)
=
a
b
+
⟨
p
(
x
)
⟩
=
ψ
(
a
b
)
ψ
a
+
⟨
p
(
x
)
⟩
=
ψ
(
a
)
=
ψ
(
b
)
=
b
+
⟨
p
(
x
)
⟩
a
−
b
p
(
x
)
⟨
p
(
x
)
⟩
p
(
x
)
a
−
b
=
0
a
=
b
ψ
ψ
F
{
a
+
⟨
p
(
x
)
⟩
:
a
∈
F
}
E
E
F
p
(
x
)
α
∈
E
α
=
x
+
⟨
p
(
x
)
⟩
α
E
p
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
p
(
α
)
=
a
0
+
a
1
(
x
+
⟨
p
(
x
)
⟩
)
+
⋯
+
a
n
(
x
+
⟨
p
(
x
)
⟩
)
n
=
a
0
+
(
a
1
x
+
⟨
p
(
x
)
⟩
)
+
⋯
+
(
a
n
x
n
+
⟨
p
(
x
)
⟩
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
+
⟨
p
(
x
)
⟩
=
0
+
⟨
p
(
x
)
⟩
α
∈
E
=
F
[
x
]
/
⟨
p
(
x
)
⟩
α
p
(
x
)
p
(
x
)
=
x
5
+
x
4
+
1
∈
Z
2
[
x
]
p
(
x
)
x
2
+
x
+
1
x
3
+
x
+
1
E
Z
2
p
(
x
)
E
E
Z
2
[
x
]
/
⟨
x
2
+
x
+
1
⟩
Z
2
[
x
]
/
⟨
x
3
+
x
+
1
⟩
Z
2
[
x
]
/
⟨
x
3
+
x
+
1
⟩
2
3
=
8
α
E
F
F
f
(
α
)
=
0
f
(
x
)
∈
F
[
x
]
E
F
F
E
F
F
E
F
E
F
α
1
,
…
,
α
n
E
F
α
1
,
…
,
α
n
F
(
α
1
,
…
,
α
n
)
F
α
1
,
…
,
α
n
E
=
F
(
α
)
α
∈
E
E
F
2
i
Q
x
2
−
2
x
2
+
1
π
e
Q
R
Q
Q
[
0
,
1
]
π
+
e
Q
C
Q
2
+
3
Q
α
=
2
+
3
α
2
=
2
+
3
α
2
−
2
=
3
(
α
2
−
2
)
2
=
3
α
4
−
4
α
2
+
1
=
0
α
x
4
−
4
x
2
+
1
∈
Q
[
x
]
E
F
E
F
E
F
α
∈
E
α
F
F
(
α
)
F
(
x
)
F
[
x
]
ϕ
α
:
F
[
x
]
→
E
α
α
F
ϕ
α
(
p
(
x
)
)
=
p
(
α
)
0
p
(
x
)
∈
F
[
x
]
ker
ϕ
α
=
{
0
}
ϕ
α
E
F
[
x
]
F
[
x
]
F
(
x
)
E
E
F
α
∈
E
α
F
p
(
x
)
∈
F
[
x
]
p
(
α
)
=
0
f
(
x
)
F
[
x
]
f
(
α
)
=
0
p
(
x
)
f
(
x
)
ϕ
α
:
F
[
x
]
→
E
ϕ
α
p
(
x
)
∈
F
[
x
]
deg
p
(
x
)
≥
1
F
[
x
]
α
⟨
p
(
x
)
⟩
F
[
x
]
α
f
(
α
)
=
0
f
(
x
)
f
(
x
)
∈
⟨
p
(
x
)
⟩
p
(
x
)
f
(
x
)
p
(
x
)
α
α
β
p
(
x
)
β
∈
F
p
(
x
)
=
r
(
x
)
s
(
x
)
p
(
x
)
p
(
α
)
=
0
r
(
α
)
s
(
α
)
=
0
r
(
α
)
=
0
s
(
α
)
=
0
p
p
(
x
)
E
F
α
∈
E
F
p
(
x
)
α
F
p
(
x
)
α
F
f
(
x
)
=
x
2
−
2
g
(
x
)
=
x
4
−
4
x
2
+
1
2
2
+
3
E
F
α
∈
E
F
F
(
α
)
≅
F
[
x
]
/
⟨
p
(
x
)
⟩
p
(
x
)
α
F
ϕ
α
:
F
[
x
]
→
E
⟨
p
(
x
)
⟩
p
(
x
)
α
ϕ
α
E
F
(
α
)
F
α
E
=
F
(
α
)
F
α
∈
E
F
α
F
n
β
∈
E
β
=
b
0
+
b
1
α
+
⋯
+
b
n
−
1
α
n
−
1
b
i
∈
F
ϕ
α
(
F
[
x
]
)
≅
F
(
α
)
E
=
F
(
α
)
ϕ
α
(
f
(
x
)
)
=
f
(
α
)
f
(
α
)
α
F
p
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
0
α
p
(
α
)
=
0
α
n
=
−
a
n
−
1
α
n
−
1
−
⋯
−
a
0
α
n
+
1
=
α
α
n
=
−
a
n
−
1
α
n
−
a
n
−
2
α
n
−
1
−
⋯
−
a
0
α
=
−
a
n
−
1
(
−
a
n
−
1
α
n
−
1
−
⋯
−
a
0
)
−
a
n
−
2
α
n
−
1
−
⋯
−
a
0
α
α
m
m
≥
n
α
n
β
∈
F
(
α
)
β
=
b
0
+
b
1
α
+
⋯
+
b
n
−
1
α
n
−
1
β
=
b
0
+
b
1
α
+
⋯
+
b
n
−
1
α
n
−
1
=
c
0
+
c
1
α
+
⋯
+
c
n
−
1
α
n
−
1
b
i
c
i
F
g
(
x
)
=
(
b
0
−
c
0
)
+
(
b
1
−
c
1
)
x
+
⋯
+
(
b
n
−
1
−
c
n
−
1
)
x
n
−
1
F
[
x
]
g
(
α
)
=
0
g
(
x
)
p
(
x
)
α
g
(
x
)
b
0
−
c
0
=
b
1
−
c
1
=
⋯
=
b
n
−
1
−
c
n
−
1
=
0
b
i
=
c
i
i
=
0
,
1
,
…
,
n
−
1
x
2
+
1
R
⟨
x
2
+
1
⟩
R
[
x
]
E
=
R
[
x
]
/
⟨
x
2
+
1
⟩
R
x
2
+
1
α
=
x
+
⟨
x
2
+
1
⟩
E
E
R
(
α
)
=
{
a
+
b
α
:
a
,
b
∈
R
}
α
2
=
−
1
E
α
2
+
1
=
(
x
+
⟨
x
2
+
1
⟩
)
2
+
(
1
+
⟨
x
2
+
1
⟩
)
=
(
x
2
+
1
)
+
⟨
x
2
+
1
⟩
=
0
R
(
α
)
C
a
+
b
α
a
+
b
i
E
F
E
F
E
F
E
E
F
E
F
F
E
=
F
(
α
)
F
{
1
,
α
,
α
2
,
…
,
α
n
−
1
}
E
F
F
n
E
n
F
[
E
:
F
]
=
n
E
F
E
F
E
F
α
∈
E
[
E
:
F
]
=
n
1
,
α
,
…
,
α
n
a
i
∈
F
a
n
α
n
+
a
n
−
1
α
n
−
1
+
⋯
+
a
1
α
+
a
0
=
0
p
(
x
)
=
a
n
x
n
+
⋯
+
a
0
∈
F
[
x
]
p
(
α
)
=
0
F
R
Q
Q
E
F
K
E
K
F
[
K
:
F
]
=
[
K
:
E
]
[
E
:
F
]
{
α
1
,
…
,
α
n
}
E
F
{
β
1
,
…
,
β
m
}
K
E
{
α
i
β
j
}
K
F
K
u
∈
K
u
=
∑
j
=
1
m
b
j
β
j
b
j
=
∑
i
=
1
n
a
i
j
α
i
b
j
∈
E
a
i
j
∈
F
u
=
∑
j
=
1
m
(
∑
i
=
1
n
a
i
j
α
i
)
β
j
=
∑
i
,
j
a
i
j
(
α
i
β
j
)
m
n
α
i
β
j
K
F
{
α
i
β
j
}
v
1
,
v
2
,
…
,
v
n
V
c
1
v
1
+
c
2
v
2
+
⋯
+
c
n
v
n
=
0
c
1
=
c
2
=
⋯
=
c
n
=
0
u
=
∑
i
,
j
c
i
j
(
α
i
β
j
)
=
0
c
i
j
∈
F
c
i
j
u
∑
j
=
1
m
(
∑
i
=
1
n
c
i
j
α
i
)
β
j
=
0
∑
i
c
i
j
α
i
∈
E
β
j
E
∑
i
=
1
n
c
i
j
α
i
=
0
j
α
j
F
c
i
j
=
0
i
j
F
i
i
=
1
,
…
,
k
F
i
+
1
F
i
F
k
F
1
[
F
k
:
F
1
]
=
[
F
k
:
F
k
−
1
]
⋯
[
F
2
:
F
1
]
E
F
α
∈
E
F
p
(
x
)
β
∈
F
(
α
)
q
(
x
)
deg
q
(
x
)
deg
p
(
x
)
deg
p
(
x
)
=
[
F
(
α
)
:
F
]
deg
q
(
x
)
=
[
F
(
β
)
:
F
]
F
⊂
F
(
β
)
⊂
F
(
α
)
[
F
(
α
)
:
F
]
=
[
F
(
α
)
:
F
(
β
)
]
[
F
(
β
)
:
F
]
Q
3
+
5
3
+
5
x
4
−
16
x
2
+
4
[
Q
(
3
+
5
)
:
Q
]
=
4
{
1
,
3
}
Q
(
3
)
Q
3
+
5
Q
(
3
)
5
Q
(
3
)
{
1
,
5
}
Q
(
3
,
5
)
=
(
Q
(
3
)
)
(
5
)
Q
(
3
)
{
1
,
3
,
5
,
3
5
=
15
}
Q
(
3
,
5
)
=
Q
(
3
+
5
)
Q
F
(
α
1
,
…
,
α
n
)
F
n
>
1
Q
(
5
3
,
5
i
)
5
5
5
3
5
5
i
∉
Q
(
5
3
)
[
Q
(
5
3
,
5
i
)
:
Q
(
5
3
)
]
=
2
{
1
,
5
i
}
Q
(
5
3
,
5
i
)
Q
(
5
3
)
{
1
,
5
3
,
(
5
3
)
2
}
Q
(
5
3
)
Q
Q
(
5
3
,
5
i
)
Q
{
1
,
5
i
,
5
3
,
(
5
3
)
2
,
(
5
6
)
5
i
,
(
5
6
)
7
i
=
5
5
6
i
or
5
6
i
}
5
6
i
x
6
+
5
Q
p
=
5
Q
⊂
Q
(
5
6
i
)
⊂
Q
(
5
3
,
5
i
)
Q
(
5
6
i
)
=
Q
(
5
3
,
5
i
)
6
E
F
E
F
α
1
,
…
,
α
n
∈
E
E
=
F
(
α
1
,
…
,
α
n
)
E
=
F
(
α
1
,
…
,
α
n
)
⊃
F
(
α
1
,
…
,
α
n
−
1
)
⊃
⋯
⊃
F
(
α
1
)
⊃
F
F
(
α
1
,
…
,
α
i
)
F
(
α
1
,
…
,
α
i
−
1
)
⇒
E
F
E
F
α
1
,
…
,
α
n
E
E
=
F
(
α
1
,
…
,
α
n
)
α
i
F
⇒
E
=
F
(
α
1
,
…
,
α
n
)
α
i
F
E
=
F
(
α
1
,
…
,
α
n
)
⊃
F
(
α
1
,
…
,
α
n
−
1
)
⊃
⋯
⊃
F
(
α
1
)
⊃
F
F
(
α
1
,
…
,
α
i
)
F
(
α
1
,
…
,
α
i
−
1
)
⇒
E
=
F
(
α
1
,
…
,
α
n
)
⊃
F
(
α
1
,
…
,
α
n
−
1
)
⊃
⋯
⊃
F
(
α
1
)
⊃
F
F
(
α
1
,
…
,
α
i
)
F
(
α
1
,
…
,
α
i
−
1
)
F
(
α
1
,
…
,
α
i
)
=
F
(
α
1
,
…
,
α
i
−
1
)
(
α
i
)
α
i
F
(
α
1
,
…
,
α
i
−
1
)
[
F
(
α
1
,
…
,
α
i
)
:
F
(
α
1
,
…
,
α
i
−
1
)
]
i
[
E
:
F
]
F
E
p
(
x
)
E
E
F
E
F
α
,
β
∈
E
F
F
(
α
,
β
)
F
F
(
α
,
β
)
F
α
±
β
α
β
α
/
β
β
0
F
E
F
Q
E
F
F
E
E
F
F
F
[
x
]
F
F
F
[
x
]
F
[
x
]
F
p
(
x
)
∈
F
[
x
]
p
(
x
)
F
α
x
−
α
p
(
x
)
p
(
x
)
=
(
x
−
α
)
q
1
(
x
)
deg
q
1
(
x
)
=
deg
p
(
x
)
−
1
q
1
(
x
)
p
(
x
)
=
(
x
−
α
)
(
x
−
β
)
q
2
(
x
)
deg
q
2
(
x
)
=
deg
p
(
x
)
−
2
p
(
x
)
p
(
x
)
F
[
x
]
a
x
−
b
p
(
b
/
a
)
=
0
F
F
E
E
F
F
⊂
E
α
∈
E
α
x
−
α
α
∈
F
F
=
E
F
F
p
(
x
)
F
[
x
]
F
p
(
x
)
E
F
p
(
x
)
F
p
(
x
)
p
(
x
)
F
p
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
F
[
x
]
E
F
p
(
x
)
α
1
,
…
,
α
n
E
E
=
F
(
α
1
,
…
,
α
n
)
p
(
x
)
=
(
x
−
α
1
)
(
x
−
α
2
)
⋯
(
x
−
α
n
)
p
(
x
)
∈
F
[
x
]
E
E
[
x
]
p
(
x
)
=
x
4
+
2
x
2
−
8
Q
[
x
]
p
(
x
)
x
2
−
2
x
2
+
4
Q
(
2
,
i
)
p
(
x
)
p
(
x
)
=
x
3
−
3
Q
[
x
]
p
(
x
)
Q
(
3
3
)
p
(
x
)
−
3
3
±
(
3
6
)
5
i
2
Q
(
3
3
)
p
(
x
)
∈
F
[
x
]
E
p
(
x
)
p
(
x
)
deg
p
(
x
)
=
1
p
(
x
)
E
=
F
k
1
≤
k
<
n
deg
p
(
x
)
=
n
p
(
x
)
K
p
(
x
)
α
1
K
p
(
x
)
=
(
x
−
α
1
)
q
(
x
)
q
(
x
)
∈
K
[
x
]
deg
q
(
x
)
=
n
−
1
E
⊃
K
q
(
x
)
α
2
,
…
,
α
n
p
(
x
)
E
=
K
(
α
2
,
…
,
α
n
)
=
F
(
α
1
,
…
,
α
n
)
p
(
x
)
K
L
p
(
x
)
∈
F
[
x
]
ϕ
:
K
→
L
F
ϕ
:
E
→
F
K
E
α
∈
K
E
p
(
x
)
L
F
β
F
[
x
]
p
(
x
)
ϕ
ϕ
ϕ
¯
:
E
(
α
)
→
F
(
β
)
ϕ
¯
(
α
)
=
β
ϕ
¯
ϕ
E
p
(
x
)
n
E
(
α
)
1
,
α
,
…
,
α
n
−
1
ϕ
¯
(
a
0
+
a
1
α
+
⋯
+
a
n
−
1
α
n
−
1
)
=
ϕ
(
a
0
)
+
ϕ
(
a
1
)
β
+
⋯
+
ϕ
(
a
n
−
1
)
β
n
−
1
a
0
+
a
1
α
+
⋯
+
a
n
−
1
α
n
−
1
E
(
α
)
ϕ
¯
ϕ
¯
ϕ
E
[
x
]
F
[
x
]
ϕ
ϕ
(
a
0
+
a
1
x
+
⋯
+
a
n
x
n
)
=
ϕ
(
a
0
)
+
ϕ
(
a
1
)
x
+
⋯
+
ϕ
(
a
n
)
x
n
ϕ
:
E
→
F
ϕ
(
p
(
x
)
)
=
q
(
x
)
ϕ
⟨
p
(
x
)
⟩
⟨
q
(
x
)
⟩
ψ
:
E
[
x
]
/
⟨
p
(
x
)
⟩
→
F
[
x
]
/
⟨
q
(
x
)
⟩
σ
:
E
[
x
]
/
⟨
p
(
x
)
⟩
→
E
(
α
)
τ
:
F
[
x
]
/
⟨
q
(
x
)
⟩
→
F
(
β
)
α
β
ϕ
¯
=
τ
ψ
σ
−
1
E
[
x
]
/
⟨
p
(
x
)
⟩
→
ψ
F
[
x
]
/
⟨
q
(
x
)
⟩
↓
σ
↓
τ
E
(
α
)
→
ϕ
¯
F
(
β
)
↓
↓
E
→
ϕ
F
ϕ
:
E
→
F
p
(
x
)
E
[
x
]
q
(
x
)
F
[
x
]
K
p
(
x
)
L
q
(
x
)
ϕ
ψ
:
K
→
L
p
(
x
)
p
(
x
)
E
q
(
x
)
F
deg
p
(
x
)
=
1
K
=
E
L
=
F
n
K
p
(
x
)
p
(
x
)
K
α
E
⊂
E
(
α
)
⊂
K
β
q
(
x
)
L
F
⊂
F
(
β
)
⊂
L
ϕ
¯
:
E
(
α
)
→
F
(
β
)
ϕ
¯
(
α
)
=
β
ϕ
¯
ϕ
E
K
→
ψ
L
↓
σ
↓
τ
E
(
α
)
→
ϕ
¯
F
(
β
)
↓
↓
E
→
ϕ
F
p
(
x
)
=
(
x
−
α
)
f
(
x
)
q
(
x
)
=
(
x
−
β
)
g
(
x
)
f
(
x
)
g
(
x
)
p
(
x
)
q
(
x
)
K
f
(
x
)
E
(
α
)
L
g
(
x
)
F
(
β
)
ψ
:
K
→
L
ψ
ϕ
¯
E
(
α
)
ψ
:
K
→
L
ψ
ϕ
E
p
(
x
)
F
[
x
]
K
p
(
x
)
30
∘
90
∘
20
∘
60
∘
α
|
α
|
F
α
β
α
+
β
α
−
β
α
β
α
/
β
β
0
α
β
α
>
β
α
+
β
α
−
β
α
β
β
>
1
△
A
B
C
△
A
D
E
α
/
1
=
x
/
β
x
α
β
β
<
1
α
/
β
β
0
α
α
△
A
B
D
△
B
C
D
△
A
B
C
1
/
x
=
x
/
α
x
2
=
α
P
=
(
p
,
q
)
p
q
F
R
F
a
x
+
b
y
+
c
=
0
a
b
c
F
F
F
x
2
+
y
2
+
d
x
+
e
y
+
f
=
0
d
e
f
F
(
x
1
,
y
1
)
(
x
2
,
y
2
)
F
x
1
=
x
2
x
−
x
1
=
0
a
x
+
b
y
+
c
=
0
x
1
x
2
y
−
y
1
=
(
y
2
−
y
1
x
2
−
x
1
)
(
x
−
x
1
)
(
x
1
,
y
1
)
r
(
x
−
x
1
)
2
+
(
y
−
y
1
)
2
−
r
2
=
0
F
R
R
F
F
F
F
R
R
F
F
R
a
x
+
b
y
+
c
=
0
F
F
x
2
+
y
2
+
d
1
x
+
e
1
y
+
f
1
=
0
x
2
+
y
2
+
d
2
x
+
e
2
y
+
f
2
=
0
d
i
e
i
f
i
F
i
=
1
,
2
x
2
+
y
2
+
d
1
x
+
e
1
x
+
f
1
=
0
(
d
1
−
d
2
)
x
+
b
(
e
2
−
e
1
)
y
+
(
f
2
−
f
1
)
=
0
a
x
+
b
y
+
c
=
0
x
2
+
y
2
+
d
x
+
e
y
+
f
=
0
y
A
x
2
+
B
x
+
C
=
0
A
B
C
F
x
x
=
−
B
±
B
2
−
4
A
C
2
A
F
(
α
)
α
=
B
2
−
4
A
C
>
0
F
F
F
(
α
)
α
F
α
Q
=
F
0
⊂
F
1
⊂
⋯
⊂
F
k
F
i
=
F
i
−
1
(
α
i
)
α
i
∈
F
i
α
∈
F
k
k
>
0
[
Q
(
α
)
:
Q
]
=
2
k
F
i
α
i
[
F
k
:
Q
]
=
[
F
k
:
F
k
−
1
]
[
F
k
−
1
:
F
k
−
2
]
⋯
[
F
1
:
Q
]
=
2
k
Q
1
1
2
2
3
2
3
x
3
−
2
Q
[
Q
(
2
3
)
:
Q
]
=
3
3
2
1
π
π
π
π
20
∘
60
∘
cos
3
θ
=
cos
(
2
θ
+
θ
)
=
cos
2
θ
cos
θ
−
sin
2
θ
sin
θ
=
(
2
cos
2
θ
−
1
)
cos
θ
−
2
sin
2
θ
cos
θ
=
(
2
cos
2
θ
−
1
)
cos
θ
−
2
(
1
−
cos
2
θ
)
cos
θ
=
4
cos
3
θ
−
3
cos
θ
θ
α
=
cos
θ
θ
=
20
∘
cos
3
θ
=
cos
60
∘
=
1
/
2
4
α
3
−
3
α
=
1
2
α
8
x
3
−
6
x
−
1
Z
[
x
]
Q
[
x
]
[
Q
(
α
)
:
Q
]
=
3
α
x
2
+
y
2
=
z
2
x
n
+
y
n
=
z
n
n
≥
3
p
(
x
,
y
)
Z
[
x
,
y
]
E
F
F
α
∈
E
E
F
α
F
p
(
x
)
∈
F
[
x
]
Q
Q
1
/
3
+
7
3
+
5
3
3
+
2
i
cos
θ
+
i
sin
θ
θ
=
2
π
/
n
n
∈
N
2
3
−
i
x
4
−
(
2
/
3
)
x
2
−
62
/
9
x
4
−
2
x
2
+
25
Q
(
3
,
6
)
Q
Q
(
2
3
,
3
3
)
Q
Q
(
2
,
i
)
Q
Q
(
3
,
5
,
7
)
Q
Q
(
2
,
2
3
)
Q
Q
(
8
)
Q
(
2
)
Q
(
i
,
2
+
i
,
3
+
i
)
Q
Q
(
2
+
5
)
Q
(
5
)
Q
(
2
,
6
+
10
)
Q
(
3
+
5
)
{
1
,
2
,
3
,
6
}
{
1
,
i
,
2
,
2
i
}
{
1
,
2
1
/
6
,
2
1
/
3
,
2
1
/
2
,
2
2
/
3
,
2
5
/
6
}
x
4
−
10
x
2
+
21
Q
x
4
+
1
Q
x
3
+
2
x
+
2
Z
3
x
3
−
3
Q
Q
(
3
,
7
)
Q
(
3
4
,
i
)
Q
Q
(
3
4
,
i
)
Q
[
Q
(
3
4
,
i
)
:
Q
]
=
8
F
Q
(
3
4
,
i
)
[
F
:
Q
]
=
2
F
Q
(
3
4
,
i
)
[
F
:
Q
]
=
4
Z
2
[
x
]
/
⟨
x
3
+
x
+
1
⟩
Z
2
[
x
]
/
⟨
x
3
+
x
+
1
⟩
α
1
+
α
α
2
1
+
α
2
α
+
α
2
1
+
α
+
α
2
α
3
+
α
+
1
=
0
9
20
cos
1
∘
Q
Q
(
3
,
3
4
,
3
8
,
…
)
Q
π
Q
(
π
3
)
p
(
x
)
n
F
[
x
]
E
p
(
x
)
[
E
:
F
]
≤
n
!
Q
(
2
)
≅
Q
(
3
)
Q
(
3
4
)
Q
(
3
4
i
)
K
E
E
F
K
F
E
F
K
E
α
∈
K
α
F
α
E
p
(
x
)
=
β
0
+
β
1
x
+
⋯
+
β
n
x
n
E
[
x
]
α
F
(
β
0
,
…
,
β
n
)
Z
[
x
]
/
⟨
x
3
−
2
⟩
F
p
p
(
x
)
=
x
p
−
a
F
F
E
F
p
(
x
)
F
[
x
]
E
p
(
x
)
F
[
x
]
F
α
β
β
0
α
/
β
R
Q
Q
E
F
σ
E
F
α
∈
E
σ
α
E
Q
(
3
,
7
)
=
Q
(
3
+
7
)
Q
(
a
,
b
)
=
Q
(
a
+
b
)
gcd
(
a
,
b
)
=
1
{
1
,
3
,
7
,
21
}
Q
(
3
,
7
)
Q
Q
(
3
,
7
)
⊃
Q
(
3
+
7
)
[
Q
(
3
,
7
)
:
Q
]
=
4
[
Q
(
3
+
7
)
:
Q
]
=
2
3
+
7
Q
(
3
,
7
)
=
Q
(
3
+
7
)
E
F
[
E
:
F
]
=
2
E
F
f
(
x
)
∈
F
[
x
]
p
(
x
)
Z
6
[
x
]
R
p
(
x
)
R
E
F
α
∈
E
[
F
(
α
)
:
F
(
α
3
)
]
α
,
β
Q
α
β
α
+
β
E
F
α
∈
E
F
F
(
α
)
F
F
β
∈
F
(
α
)
F
β
=
p
(
α
)
/
q
(
α
)
p
q
α
q
(
α
)
0
F
β
F
f
(
x
)
∈
F
[
x
]
f
(
β
)
=
0
f
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
0
=
f
(
β
)
=
f
(
p
(
α
)
q
(
α
)
)
=
a
0
+
a
1
(
p
(
α
)
q
(
α
)
)
+
⋯
+
a
n
(
p
(
α
)
q
(
α
)
)
n
q
(
α
)
n
F
[
x
]
α
α
p
(
x
)
∈
F
[
x
]
deg
p
=
n
[
F
(
α
)
:
F
]
=
n
Q
⊂
Q
[
3
]
⊂
Q
[
3
,
2
]
2
−
3
2
3
Q
p
(
x
)
=
x
4
+
x
2
−
1
a
2
+
1
8
8
p
(
x
)
=
x
4
+
x
2
−
1
(
w
−
r
)
60
20
2
p
(
x
)
=
x
5
+
2
x
4
+
1
Z
3
p
(
x
)
Z
3
p
(
x
)
3
5
p
(
x
)
p
(
x
)
3
5
=
243
p
(
x
)
Z
3
p
(
x
)
p
(
x
)
r
(
x
)
=
x
4
+
2
x
+
2
p
=
2
r
(
x
)
r
(
x
)
s
(
x
)
=
x
4
+
x
2
+
1
s
(
x
)
K
q
(
x
)
=
x
3
+
3
x
2
+
3
x
−
2
q
(
x
)
K
K
K
q
(
x
)
L
M
q
(
x
)
P
q
(
x
)
P
q
(
x
)
Z
p
p
p
n
p
n
F
p
p
α
F
p
α
=
0
F
0
p
F
n
n
α
=
0
α
F
F
p
p
n
F
F
p
p
p
F
p
F
p
n
n
∈
N
ϕ
:
Z
→
F
ϕ
(
n
)
=
n
⋅
1
F
p
ϕ
p
Z
ϕ
F
Z
p
K
F
K
K
[
F
:
K
]
=
n
F
F
K
α
1
,
…
,
α
n
∈
F
α
F
α
=
a
1
α
1
+
⋯
+
a
n
α
n
a
i
K
p
K
p
n
α
i
F
p
n
p
D
p
a
p
n
+
b
p
n
=
(
a
+
b
)
p
n
n
n
n
=
1
(
a
+
b
)
p
=
∑
k
=
0
p
(
p
k
)
a
k
b
p
−
k
0
<
k
<
p
(
p
k
)
=
p
!
k
!
(
p
−
k
)
!
p
p
k
!
(
p
−
k
)
!
D
p
(
a
+
b
)
p
=
a
p
+
b
p
k
1
≤
k
≤
n
(
a
+
b
)
p
n
+
1
=
(
(
a
+
b
)
p
)
p
n
=
(
a
p
+
b
p
)
p
n
=
(
a
p
)
p
n
+
(
b
p
)
p
n
=
a
p
n
+
1
+
b
p
n
+
1
n
+
1
F
f
(
x
)
∈
F
[
x
]
n
n
f
(
x
)
f
(
x
)
f
E
F
F
E
F
[
x
]
x
2
−
2
Q
(
x
−
2
)
(
x
+
2
)
Q
(
2
)
Q
α
=
a
+
b
2
Q
(
2
)
b
=
0
α
x
−
a
b
0
α
x
2
−
2
a
x
+
a
2
−
2
b
2
=
(
x
−
(
a
+
b
2
)
)
(
x
−
(
a
−
b
2
)
)
f
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
F
[
x
]
f
(
x
)
f
′
(
x
)
=
a
1
+
2
a
2
x
+
⋯
+
n
a
n
x
n
−
1
F
f
(
x
)
∈
F
[
x
]
f
(
x
)
f
(
x
)
f
′
(
x
)
f
(
x
)
f
(
x
)
F
f
(
x
)
=
(
x
−
α
1
)
(
x
−
α
2
)
⋯
(
x
−
α
n
)
α
i
α
j
i
j
f
(
x
)
f
′
(
x
)
=
(
x
−
α
2
)
⋯
(
x
−
α
n
)
+
(
x
−
α
1
)
(
x
−
α
3
)
⋯
(
x
−
α
n
)
+
⋯
+
(
x
−
α
1
)
⋯
(
x
−
α
n
−
1
)
f
(
x
)
f
′
(
x
)
f
(
x
)
=
(
x
−
α
)
k
g
(
x
)
k
>
1
f
′
(
x
)
=
k
(
x
−
α
)
k
−
1
g
(
x
)
+
(
x
−
α
)
k
g
′
(
x
)
f
(
x
)
f
′
(
x
)
p
n
F
p
n
p
n
x
p
n
−
x
Z
p
f
(
x
)
=
x
p
n
−
x
F
f
(
x
)
f
(
x
)
p
n
F
f
′
(
x
)
=
p
n
x
p
n
−
1
−
1
=
−
1
f
(
x
)
f
(
x
)
F
f
(
x
)
α
β
f
(
x
)
α
+
β
α
β
f
(
x
)
α
p
n
+
β
p
n
=
(
α
+
β
)
p
n
α
p
n
β
p
n
=
(
α
β
)
p
n
f
(
x
)
f
(
x
)
α
f
(
x
)
−
α
f
(
x
)
f
(
−
α
)
=
(
−
α
)
p
n
−
(
−
α
)
=
−
α
p
n
+
α
=
−
(
α
p
n
−
α
)
=
0
p
p
=
2
f
(
−
α
)
=
(
−
α
)
2
n
−
(
−
α
)
=
α
+
α
=
0
α
0
(
α
−
1
)
p
n
=
(
α
p
n
)
−
1
=
α
−
1
f
(
x
)
F
f
(
x
)
F
E
p
n
E
F
E
f
(
x
)
f
(
x
)
α
E
E
p
n
−
1
α
p
n
−
1
=
1
α
p
n
−
α
=
0
E
p
n
E
f
(
x
)
p
n
p
n
GF
(
p
n
)
p
n
GF
(
p
n
)
p
m
m
n
m
∣
n
m
>
0
GF
(
p
n
)
GF
(
p
m
)
F
E
=
GF
(
p
n
)
F
K
p
m
K
Z
p
m
∣
n
[
E
:
K
]
=
[
E
:
F
]
[
F
:
K
]
m
∣
n
m
>
0
p
m
−
1
p
n
−
1
x
p
m
−
1
−
1
x
p
n
−
1
−
1
x
p
m
−
x
x
p
n
−
x
x
p
m
−
x
x
p
n
−
x
GF
(
p
n
)
x
p
m
−
x
GF
(
p
m
)
GF
(
p
24
)
GF
(
p
24
)
F
F
F
∗
F
G
F
∗
F
G
G
F
∗
n
G
≅
Z
p
1
e
1
×
⋯
×
Z
p
k
e
k
n
=
p
1
e
1
⋯
p
k
e
k
p
1
,
…
,
p
k
m
p
1
e
1
,
…
,
p
k
e
k
G
m
α
G
x
r
−
1
r
m
α
x
m
−
1
x
m
−
1
m
F
n
≤
m
m
≤
|
G
|
m
=
n
G
n
E
F
F
α
E
∗
E
E
=
F
(
α
)
GF
(
2
4
)
Z
2
/
⟨
1
+
x
+
x
4
⟩
GF
(
2
4
)
{
a
0
+
a
1
α
+
a
2
α
2
+
a
3
α
3
:
a
i
∈
Z
2
and
1
+
α
+
α
4
=
0
}
1
+
α
+
α
4
=
0
GF
(
2
4
)
GF
(
2
4
)
Z
15
α
α
1
=
α
α
6
=
α
2
+
α
3
α
11
=
α
+
α
2
+
α
3
α
2
=
α
2
α
7
=
1
+
α
+
α
3
α
12
=
1
+
α
+
α
2
+
α
3
α
3
=
α
3
α
8
=
1
+
α
2
α
13
=
1
+
α
2
+
α
3
α
4
=
1
+
α
α
9
=
α
+
α
3
α
14
=
1
+
α
3
α
5
=
α
+
α
2
α
10
=
1
+
α
+
α
2
α
15
=
1.
(
n
,
k
)
E
:
Z
2
k
→
Z
2
n
D
:
Z
2
n
→
Z
2
k
D
H
∈
M
k
×
n
(
Z
2
)
ϕ
:
Z
2
k
→
Z
2
n
(
n
,
k
)
ϕ
(
a
1
,
a
2
,
…
,
a
n
)
n
(
a
n
,
a
1
,
a
2
,
…
,
a
n
−
1
)
(
6
,
3
)
G
1
=
(
1
0
0
0
1
0
0
0
1
1
0
0
0
1
0
0
0
1
)
and
G
2
=
(
1
0
0
1
1
0
1
1
1
1
1
1
0
1
1
0
0
1
)
(
000
)
↦
(
000000
)
(
100
)
↦
(
100100
)
(
001
)
↦
(
001001
)
(
101
)
↦
(
101101
)
(
010
)
↦
(
010010
)
(
110
)
↦
(
110110
)
(
011
)
↦
(
011011
)
(
111
)
↦
(
111111
)
.
(
000
)
↦
(
000000
)
(
100
)
↦
(
111100
)
(
001
)
↦
(
001111
)
(
101
)
↦
(
110011
)
(
010
)
↦
(
011110
)
(
110
)
↦
(
100010
)
(
011
)
↦
(
010001
)
(
111
)
↦
(
101101
)
.
(
101101
)
(
011011
)
Z
2
n
Z
2
[
x
]
n
(
a
0
,
a
1
,
…
,
a
n
−
1
)
f
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
−
1
x
n
−
1
f
(
x
)
n
−
1
5
(
10011
)
1
+
0
x
+
0
x
2
+
1
x
3
+
1
x
4
=
1
+
x
3
+
x
4
f
(
x
)
∈
Z
2
[
x
]
deg
f
(
x
)
<
n
n
x
+
x
2
+
x
4
5
(
01101
)
g
(
x
)
Z
2
[
x
]
n
−
k
(
n
,
k
)
C
(
a
0
,
…
,
a
k
−
1
)
k
f
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
k
−
1
x
k
−
1
Z
2
[
x
]
f
(
x
)
g
(
x
)
C
Z
2
[
x
]
n
g
(
x
)
g
(
x
)
=
1
+
x
3
(
6
,
3
)
C
3
(
a
0
,
a
1
,
a
2
)
f
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
1
+
x
3
ϕ
:
Z
2
3
→
Z
2
6
ϕ
:
f
(
x
)
↦
g
(
x
)
f
(
x
)
Z
2
n
Z
2
ϕ
ϕ
ϕ
(
a
0
,
a
1
,
a
2
)
=
(
000000
)
0
+
0
x
+
0
x
2
+
0
x
3
+
0
x
4
+
0
x
5
=
(
1
+
x
3
)
(
a
0
+
a
1
x
+
a
2
x
2
)
=
a
0
+
a
1
x
+
a
2
x
2
+
a
0
x
3
+
a
1
x
4
+
a
2
x
5
a
0
+
a
1
x
+
a
2
x
2
ker
ϕ
=
{
(
000
)
}
ϕ
C
1
x
x
2
(
1
+
x
3
)
⋅
1
=
1
+
x
3
(
1
+
x
3
)
x
=
x
+
x
4
(
1
+
x
3
)
x
2
=
x
2
+
x
5
G
1
H
=
(
1
0
0
1
0
0
0
1
0
0
1
0
0
0
1
0
0
1
)
2
x
n
−
1
=
(
x
−
1
)
(
x
n
−
1
+
⋯
+
x
+
1
)
R
n
=
Z
2
[
x
]
/
⟨
x
n
−
1
⟩
f
(
t
)
=
a
0
+
a
1
t
+
⋯
+
a
n
−
1
t
n
−
1
t
n
=
1
Z
2
n
R
n
Z
2
n
Z
[
x
]
/
⟨
x
n
−
1
⟩
Z
[
x
]
/
⟨
x
n
−
1
⟩
n
f
(
t
)
=
a
0
+
a
1
t
+
⋯
+
a
n
−
1
t
n
−
1
R
n
t
f
(
t
)
=
a
n
−
1
+
a
0
t
+
⋯
+
a
n
−
2
t
n
−
1
f
(
t
)
t
R
n
C
Z
2
n
R
n
=
Z
[
x
]
/
⟨
x
n
−
1
⟩
C
f
(
t
)
C
t
f
(
t
)
C
t
k
f
(
t
)
C
k
∈
N
C
f
(
t
)
,
t
f
(
t
)
,
t
2
f
(
t
)
,
…
,
t
n
−
1
f
(
t
)
p
(
t
)
p
(
t
)
f
(
t
)
C
C
C
Z
2
[
x
]
/
⟨
x
n
+
1
⟩
f
(
t
)
=
a
0
+
a
1
t
+
⋯
+
a
n
−
1
t
n
−
1
C
t
f
(
t
)
C
(
a
1
,
…
,
a
n
−
1
,
a
0
)
C
R
n
Z
2
n
R
n
ϕ
:
Z
2
[
x
]
→
R
n
ϕ
[
f
(
x
)
]
=
f
(
t
)
ϕ
x
n
−
1
C
R
n
ϕ
(
I
)
I
Z
2
[
x
]
⟨
x
n
−
1
⟩
I
Z
2
[
x
]
Z
2
I
=
⟨
g
(
x
)
⟩
Z
2
[
x
]
⟨
x
n
−
1
⟩
I
g
(
x
)
x
n
−
1
C
R
n
C
=
⟨
g
(
t
)
⟩
=
{
f
(
t
)
g
(
t
)
:
f
(
t
)
∈
R
n
and
g
(
x
)
∣
(
x
n
−
1
)
in
Z
2
[
x
]
}
C
C
x
7
−
1
x
7
−
1
=
(
1
+
x
)
(
1
+
x
+
x
3
)
(
1
+
x
2
+
x
3
)
g
(
t
)
=
(
1
+
t
+
t
3
)
C
R
7
(
7
,
4
)
g
(
t
)
t
t
2
t
3
C
G
=
(
1
0
0
0
1
1
0
0
0
1
1
0
1
0
1
1
0
1
0
1
0
0
1
0
0
0
0
1
)
(
n
,
k
)
C
t
k
x
n
−
1
=
g
(
x
)
h
(
x
)
Z
2
[
x
]
g
(
x
)
=
g
0
+
g
1
x
+
⋯
+
g
n
−
k
x
n
−
k
h
(
x
)
=
h
0
+
h
1
x
+
⋯
+
h
k
x
k
n
×
k
G
=
(
g
0
0
⋯
0
g
1
g
0
⋯
0
⋮
⋮
⋱
⋮
g
n
−
k
g
n
−
k
−
1
⋯
g
0
0
g
n
−
k
⋯
g
1
⋮
⋮
⋱
⋮
0
0
⋯
g
n
−
k
)
C
g
(
t
)
C
(
n
−
k
)
×
n
H
=
(
0
⋯
0
0
h
k
⋯
h
0
0
⋯
0
h
k
⋯
h
0
0
⋯
⋯
⋯
⋯
⋯
⋯
⋯
h
k
⋯
h
0
0
0
⋯
0
)
C
=
⟨
g
(
t
)
⟩
R
n
x
n
−
1
=
g
(
x
)
h
(
x
)
G
H
C
H
G
=
0
x
7
−
1
=
g
(
x
)
h
(
x
)
=
(
1
+
x
+
x
3
)
(
1
+
x
+
x
2
+
x
4
)
H
=
(
0
0
1
0
1
1
1
0
1
0
1
1
1
0
1
0
1
1
1
0
0
)
α
1
,
…
,
α
n
F
n
×
n
(
1
1
⋯
1
α
1
α
2
⋯
α
n
α
1
2
α
2
2
⋯
α
n
2
⋮
⋮
⋱
⋮
α
1
n
−
1
α
2
n
−
1
⋯
α
n
n
−
1
)
α
1
,
…
,
α
n
F
n
≥
2
det
(
1
1
⋯
1
α
1
α
2
⋯
α
n
α
1
2
α
2
2
⋯
α
n
2
⋮
⋮
⋱
⋮
α
1
n
−
1
α
2
n
−
1
⋯
α
n
n
−
1
)
=
∏
1
≤
j
<
i
≤
n
(
α
i
−
α
j
)
α
i
n
n
=
2
α
2
−
α
1
n
−
1
p
(
x
)
p
(
x
)
=
det
(
1
1
⋯
1
1
α
1
α
2
⋯
α
n
−
1
x
α
1
2
α
2
2
⋯
α
n
−
1
2
x
2
⋮
⋮
⋱
⋮
⋮
α
1
n
−
1
α
2
n
−
1
⋯
α
n
−
1
n
−
1
x
n
−
1
)
p
(
x
)
n
−
1
p
(
x
)
α
1
,
…
,
α
n
−
1
p
(
x
)
=
(
x
−
α
1
)
(
x
−
α
2
)
⋯
(
x
−
α
n
−
1
)
β
β
=
(
−
1
)
n
+
n
det
(
1
1
⋯
1
α
1
α
2
⋯
α
n
−
1
α
1
2
α
2
2
⋯
α
n
−
1
2
⋮
⋮
⋱
⋮
α
1
n
−
2
α
2
n
−
2
⋯
α
n
−
1
n
−
2
)
β
=
(
−
1
)
n
+
n
∏
1
≤
j
<
i
≤
n
−
1
(
α
i
−
α
j
)
x
=
α
n
C
=
⟨
g
(
t
)
⟩
R
n
ω
n
Z
2
s
ω
g
(
x
)
C
s
+
1
g
(
ω
r
)
=
g
(
ω
r
+
1
)
=
⋯
=
g
(
ω
r
+
s
−
1
)
=
0
f
(
x
)
C
s
f
(
x
)
=
a
i
0
x
i
0
+
a
i
1
x
i
1
+
⋯
+
a
i
s
−
1
x
i
s
−
1
C
a
i
g
(
ω
r
)
=
g
(
ω
r
+
1
)
=
⋯
=
g
(
ω
r
+
s
−
1
)
=
0
g
(
x
)
f
(
x
)
f
(
ω
r
)
=
f
(
ω
r
+
1
)
=
⋯
=
f
(
ω
r
+
s
−
1
)
=
0
a
i
0
(
ω
r
)
i
0
+
a
i
1
(
ω
r
)
i
1
+
⋯
+
a
i
s
−
1
(
ω
r
)
i
s
−
1
=
0
a
i
0
(
ω
r
+
1
)
i
0
+
a
i
1
(
ω
r
+
1
)
i
2
+
⋯
+
a
i
s
−
1
(
ω
r
+
1
)
i
s
−
1
=
0
⋮
a
i
0
(
ω
r
+
s
−
1
)
i
0
+
a
i
1
(
ω
r
+
s
−
1
)
i
1
+
⋯
+
a
i
s
−
1
(
ω
r
+
s
−
1
)
i
s
−
1
=
0
(
a
i
0
,
a
i
1
,
…
,
a
i
s
−
1
)
(
ω
i
0
)
r
x
0
+
(
ω
i
1
)
r
x
1
+
⋯
+
(
ω
i
s
−
1
)
r
x
n
−
1
=
0
(
ω
i
0
)
r
+
1
x
0
+
(
ω
i
1
)
r
+
1
x
1
+
⋯
+
(
ω
i
s
−
1
)
r
+
1
x
n
−
1
=
0
⋮
(
ω
i
0
)
r
+
s
−
1
x
0
+
(
ω
i
1
)
r
+
s
−
1
x
1
+
⋯
+
(
ω
i
s
−
1
)
r
+
s
−
1
x
n
−
1
=
0
(
(
ω
i
0
)
r
(
ω
i
1
)
r
⋯
(
ω
i
s
−
1
)
r
(
ω
i
0
)
r
+
1
(
ω
i
1
)
r
+
1
⋯
(
ω
i
s
−
1
)
r
+
1
⋮
⋮
⋱
⋮
(
ω
i
0
)
r
+
s
−
1
(
ω
i
1
)
r
+
s
−
1
⋯
(
ω
i
s
−
1
)
r
+
s
−
1
)
a
i
0
=
a
i
1
=
⋯
=
a
i
s
−
1
=
0
231
24
231
+
24
=
255
=
2
8
−
1
(
255
,
231
)
1
16
d
=
2
r
+
1
r
≥
0
ω
n
Z
2
m
i
(
x
)
Z
2
ω
i
g
(
x
)
=
lcm
[
m
1
(
x
)
,
m
2
(
x
)
,
…
,
m
2
r
(
x
)
]
⟨
g
(
t
)
⟩
R
n
n
d
C
d
C
=
⟨
g
(
t
)
⟩
R
n
C
d
f
(
t
)
C
f
(
ω
i
)
=
0
1
≤
i
<
d
H
=
(
1
ω
ω
2
⋯
ω
n
−
1
1
ω
2
ω
4
⋯
ω
(
n
−
1
)
(
2
)
1
ω
3
ω
6
⋯
ω
(
n
−
1
)
(
3
)
⋮
⋮
⋮
⋱
⋮
1
ω
2
r
ω
4
r
⋯
ω
(
n
−
1
)
(
2
r
)
)
C
⇒
f
(
t
)
C
g
(
x
)
∣
f
(
x
)
Z
2
[
x
]
i
=
1
,
…
,
2
r
f
(
ω
i
)
=
0
g
(
ω
i
)
=
0
f
(
ω
i
)
=
0
1
≤
i
≤
d
f
(
x
)
m
i
(
x
)
m
i
(
x
)
ω
i
g
(
x
)
∣
f
(
x
)
g
(
x
)
f
(
x
)
⇒
f
(
t
)
=
a
0
+
a
1
t
+
⋯
+
a
n
−
1
v
t
n
−
1
R
n
n
Z
2
n
x
=
(
a
0
a
1
⋯
a
n
−
1
)
t
H
x
=
(
a
0
+
a
1
ω
+
⋯
+
a
n
−
1
ω
n
−
1
a
0
+
a
1
ω
2
+
⋯
+
a
n
−
1
(
ω
2
)
n
−
1
⋮
a
0
+
a
1
ω
2
r
+
⋯
+
a
n
−
1
(
ω
2
r
)
n
−
1
)
=
(
f
(
ω
)
f
(
ω
2
)
⋮
f
(
ω
2
r
)
)
=
0
f
(
t
)
C
H
C
⇒
f
(
t
)
=
a
0
+
a
1
t
+
⋯
+
a
n
−
1
t
n
−
1
C
f
(
ω
i
)
=
0
i
=
1
,
…
,
2
r
g
(
t
)
=
lcm
[
m
1
(
t
)
,
…
,
m
2
r
(
t
)
]
C
=
⟨
g
(
t
)
⟩
x
15
−
1
∈
Z
2
[
x
]
x
15
−
1
=
(
x
+
1
)
(
x
2
+
x
+
1
)
(
x
4
+
x
+
1
)
(
x
4
+
x
3
+
1
)
(
x
4
+
x
3
+
x
2
+
x
+
1
)
ω
1
+
x
+
x
4
GF
(
2
4
)
{
a
0
+
a
1
ω
+
a
2
ω
2
+
a
3
ω
3
:
a
i
∈
Z
2
and
1
+
ω
+
ω
4
=
0
}
ω
15
ω
m
1
(
x
)
=
1
+
x
+
x
4
ω
2
ω
4
m
1
(
x
)
ω
3
m
2
(
x
)
=
1
+
x
+
x
2
+
x
3
+
x
4
g
(
x
)
=
m
1
(
x
)
m
2
(
x
)
=
1
+
x
4
+
x
6
+
x
7
+
x
8
ω
ω
2
ω
3
ω
4
m
1
(
x
)
m
2
(
x
)
x
15
−
1
(
15
,
7
)
x
15
−
1
=
g
(
x
)
h
(
x
)
h
(
x
)
=
1
+
x
4
+
x
6
+
x
7
(
0
0
0
0
0
0
0
1
1
0
1
0
0
0
1
0
0
0
0
0
0
1
1
0
1
0
0
0
1
0
0
0
0
0
0
1
1
0
1
0
0
0
1
0
0
0
0
0
0
1
1
0
1
0
0
0
1
0
0
0
0
0
0
1
1
0
1
0
0
0
1
0
0
0
0
0
0
1
1
0
1
0
0
0
1
0
0
0
0
0
0
1
1
0
1
0
0
0
1
0
0
0
0
0
0
1
1
0
1
0
0
0
1
0
0
0
0
0
0
0
)
[
GF
(
3
6
)
:
GF
(
3
3
)
]
[
GF
(
128
)
:
GF
(
16
)
]
[
GF
(
625
)
:
GF
(
25
)
]
[
GF
(
p
12
)
:
GF
(
p
2
)
]
[
GF
(
p
m
)
:
GF
(
p
n
)
]
n
∣
m
GF
(
p
30
)
α
x
3
+
x
2
+
1
Z
2
8
x
3
+
x
2
+
1
Z
2
(
α
)
Z
2
(
α
)
x
3
+
x
2
+
1
α
27
p
(
x
)
Z
3
[
x
]
3
Z
3
[
x
]
/
⟨
p
(
x
)
⟩
27
Q
∗
Z
2
[
x
]
x
5
−
1
x
6
+
x
5
+
x
4
+
x
3
+
x
2
+
x
+
1
x
9
−
1
x
4
+
x
3
+
x
2
+
x
+
1
x
5
−
1
=
(
x
+
1
)
(
x
4
+
x
3
+
x
2
+
x
+
1
)
x
9
−
1
=
(
x
+
1
)
(
x
2
+
x
+
1
)
(
x
6
+
x
3
+
1
)
Z
2
[
x
]
/
⟨
x
3
+
x
+
1
⟩
≅
Z
2
[
x
]
/
⟨
x
3
+
x
2
+
1
⟩
n
n
=
6
,
7
,
8
,
10
⟨
t
+
1
⟩
R
n
Z
2
n
7
15
x
7
−
1
=
(
x
+
1
)
(
x
3
+
x
+
1
)
(
x
3
+
x
2
+
1
)
p
Z
p
(
x
)
p
D
p
(
a
−
b
)
p
n
=
a
p
n
−
b
p
n
a
,
b
∈
D
E
F
K
E
F
E
=
F
F
⊂
E
⊂
K
K
F
K
E
p
(
x
)
∈
F
[
x
]
p
(
x
)
∈
E
[
x
]
E
F
F
q
α
∈
E
F
n
F
(
α
)
q
n
α
F
n
β
∈
F
(
α
)
β
=
a
0
+
a
1
α
+
⋯
+
a
n
−
1
α
n
−
1
a
i
∈
F
q
n
n
(
a
0
,
a
1
,
…
,
a
n
−
1
)
F
E
F
α
∈
E
E
=
F
(
α
)
n
n
Z
p
[
x
]
Φ
:
GF
(
p
n
)
→
GF
(
p
n
)
Φ
:
α
↦
α
p
n
GF
(
p
n
)
a
p
a
∈
GF
(
p
n
)
E
F
GF
(
p
n
)
|
E
|
=
p
r
|
F
|
=
p
s
E
∩
F
p
(
p
−
1
)
!
≡
−
1
(
mod
p
)
x
p
−
1
−
1
Z
p
g
(
t
)
C
R
n
g
(
x
)
1
C
(
n
,
k
)
n
−
k
R
n
Z
2
n
C
R
n
g
(
t
)
⟨
f
(
t
)
⟩
R
n
⟨
g
(
t
)
⟩
⊂
⟨
f
(
t
)
⟩
f
(
x
)
g
(
x
)
Z
2
[
x
]
C
=
⟨
g
(
t
)
⟩
R
n
x
n
−
1
=
g
(
x
)
h
(
x
)
g
(
x
)
=
g
0
+
g
1
x
+
⋯
+
g
n
−
k
x
n
−
k
h
(
x
)
=
h
0
+
h
1
x
+
⋯
+
h
k
x
k
G
n
×
k
G
=
(
g
0
0
⋯
0
g
1
g
0
⋯
0
⋮
⋮
⋱
⋮
g
n
−
k
g
n
−
k
−
1
⋯
g
0
0
g
n
−
k
⋯
g
1
⋮
⋮
⋱
⋮
0
0
⋯
g
n
−
k
)
H
(
n
−
k
)
×
n
H
=
(
0
⋯
0
0
h
k
⋯
h
0
0
⋯
0
h
k
⋯
h
0
0
⋯
⋯
⋯
⋯
⋯
⋯
⋯
h
k
⋯
h
0
0
0
⋯
0
)
G
C
H
C
H
G
=
0
C
R
n
c
(
t
)
=
c
0
+
c
1
t
+
⋯
+
c
n
−
1
t
n
−
1
w
(
t
)
=
w
0
+
w
1
t
+
⋯
w
n
−
1
t
n
−
1
R
n
a
1
,
…
,
a
k
w
(
t
)
=
c
(
t
)
+
e
(
t
)
e
(
t
)
=
t
a
1
+
t
a
2
+
⋯
+
t
a
k
a
i
c
(
t
)
w
(
t
)
a
i
w
(
t
)
w
(
ω
i
)
=
s
i
i
=
1
,
…
,
2
r
ω
n
Z
2
w
(
t
)
s
1
,
…
,
s
2
r
w
(
t
)
s
i
=
0
i
s
i
=
w
(
ω
i
)
=
e
(
ω
i
)
=
ω
i
a
1
+
ω
i
a
2
+
⋯
+
ω
i
a
k
i
=
1
,
…
,
2
r
s
(
x
)
=
(
x
+
ω
a
1
)
(
x
+
ω
a
2
)
⋯
(
x
+
ω
a
k
)
(
15
,
7
)
a
1
a
2
s
(
x
)
=
(
x
+
ω
a
1
)
(
x
+
ω
a
2
)
s
(
x
)
=
x
2
+
s
1
x
+
(
s
1
2
+
s
3
s
1
)
w
(
t
)
=
1
+
t
2
+
t
4
+
t
5
+
t
7
+
t
12
+
t
13
5
2
p
(
x
)
=
x
25
−
x
2
7
0
3
6
3
2
2
|
6
8
8
9
Q
(
3
,
7
)
x
2
−
3
x
2
−
7
3
7
Q
(
3
,
7
)
p
=
2
,
3
,
5
,
7
3
≤
n
≤
10
n
τ
E
K
=
{
b
∈
E
∣
τ
(
b
)
=
b
}
E
τ
E
E
=
G
F
(
3
6
)
p
=
2
F
a
∈
F
{
x
2
|
x
∈
F
}
{
a
−
x
2
|
x
∈
F
}
S
e
t
(
)
a
p
=
2
x
5
−
1
=
0
x
6
−
x
3
−
6
=
0
a
x
5
+
b
x
4
+
c
x
3
+
d
x
2
+
e
x
+
f
=
0
F
σ
τ
F
σ
τ
σ
−
1
F
E
F
E
F
σ
:
E
→
E
σ
(
α
)
=
α
α
∈
F
E
F
E
σ
τ
E
σ
(
α
)
=
α
τ
(
α
)
=
α
α
∈
F
σ
τ
(
α
)
=
σ
(
α
)
=
α
σ
−
1
(
α
)
=
α
E
E
F
E
E
F
E
Aut
(
E
)
E
F
E
F
G
(
E
/
F
)
=
{
σ
∈
Aut
(
E
)
:
σ
(
α
)
=
α
for all
α
∈
F
}
f
(
x
)
F
[
x
]
E
f
(
x
)
F
f
(
x
)
G
(
E
/
F
)
E
F
σ
:
a
+
b
i
↦
a
−
b
i
σ
(
a
)
=
σ
(
a
+
0
i
)
=
a
−
0
i
=
a
G
(
C
/
R
)
Q
⊂
Q
(
5
)
⊂
Q
(
3
,
5
)
a
,
b
∈
Q
(
5
)
σ
(
a
+
b
3
)
=
a
−
b
3
Q
(
3
,
5
)
Q
(
5
)
τ
(
a
+
b
5
)
=
a
−
b
5
Q
(
3
,
5
)
Q
(
3
)
μ
=
σ
τ
3
5
{
i
d
,
σ
,
τ
,
μ
}
Q
(
3
,
5
)
Q
Z
2
×
Z
2
i
d
σ
τ
μ
i
d
i
d
σ
τ
μ
σ
σ
i
d
μ
τ
τ
τ
μ
i
d
σ
μ
μ
τ
σ
i
d
Q
(
3
,
5
)
Q
{
1
,
3
,
5
,
15
}
|
G
(
Q
(
3
,
5
)
/
Q
)
|
=
[
Q
(
3
,
5
)
:
Q
)
]
=
4
E
F
f
(
x
)
F
[
x
]
G
(
E
/
F
)
f
(
x
)
E
f
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
+
⋯
+
a
n
x
n
α
∈
E
f
(
x
)
σ
∈
G
(
E
/
F
)
0
=
σ
(
0
)
=
σ
(
f
(
α
)
)
=
σ
(
a
0
+
a
1
α
+
a
2
α
2
+
⋯
+
a
n
α
n
)
=
a
0
+
a
1
σ
(
α
)
+
a
2
[
σ
(
α
)
]
2
+
⋯
+
a
n
[
σ
(
α
)
]
n
;
σ
(
α
)
f
(
x
)
E
F
α
,
β
∈
E
F
Q
(
2
)
2
−
2
Q
x
2
−
2
α
β
F
σ
:
F
(
α
)
→
F
(
β
)
σ
F
f
(
x
)
F
[
x
]
E
f
(
x
)
F
f
(
x
)
|
G
(
E
/
F
)
|
=
[
E
:
F
]
[
E
:
F
]
[
E
:
F
]
=
1
E
=
F
[
E
:
F
]
>
1
f
(
x
)
=
p
(
x
)
q
(
x
)
p
(
x
)
d
d
>
1
f
(
x
)
F
[
E
:
F
]
=
1
α
p
(
x
)
ϕ
:
F
(
α
)
→
E
ϕ
(
α
)
=
β
p
(
x
)
ϕ
:
F
(
α
)
→
F
(
β
)
f
(
x
)
p
(
x
)
d
β
∈
E
d
ϕ
:
F
(
α
)
→
F
(
β
i
)
F
β
1
,
…
,
β
d
p
(
x
)
E
→
ψ
E
↓
↓
F
(
α
)
→
ϕ
F
(
β
)
↓
↓
F
→
identity
F
E
f
(
x
)
F
F
(
α
)
E
f
(
x
)
F
(
β
)
[
E
:
F
(
α
)
]
=
[
E
:
F
]
/
d
d
ϕ
[
E
:
F
]
/
d
ψ
:
E
→
E
[
E
:
F
]
F
σ
F
σ
F
(
α
)
ϕ
ϕ
:
F
(
α
)
→
F
(
β
)
F
E
[
E
:
F
]
=
k
G
(
E
/
F
)
k
p
E
F
E
F
p
m
p
n
n
k
=
m
E
x
p
m
−
x
p
E
x
p
m
−
x
F
|
G
(
E
/
F
)
|
=
k
G
(
E
/
F
)
G
(
E
/
F
)
σ
:
E
→
E
σ
(
α
)
=
α
p
n
σ
G
(
E
/
F
)
σ
Aut
(
E
)
α
β
E
σ
(
α
+
β
)
=
(
α
+
β
)
p
n
=
α
p
n
+
β
p
n
=
σ
(
α
)
+
σ
(
β
)
σ
(
α
β
)
=
σ
(
α
)
σ
(
β
)
σ
E
σ
G
(
E
/
F
)
F
x
p
n
−
x
p
σ
F
σ
k
σ
k
(
α
)
=
α
p
n
k
=
α
p
m
=
α
G
(
E
/
F
)
σ
r
1
≤
r
<
k
x
p
n
r
−
x
p
m
Q
(
3
,
5
)
Q
Z
2
×
Z
2
H
=
{
i
d
,
σ
,
τ
,
μ
}
G
(
Q
(
3
,
5
)
/
Q
)
H
G
(
Q
(
3
,
5
)
/
Q
)
|
H
|
=
[
Q
(
3
,
5
)
:
Q
]
=
|
G
(
Q
(
3
,
5
)
/
Q
)
|
=
4
f
(
x
)
=
x
4
+
x
3
+
x
2
+
x
+
1
Q
f
(
x
)
(
x
−
1
)
f
(
x
)
=
x
5
−
1
f
(
x
)
ω
i
i
=
1
,
…
,
4
ω
=
cos
(
2
π
/
5
)
+
i
sin
(
2
π
/
5
)
f
(
x
)
Q
(
ω
)
σ
i
Q
(
ω
)
σ
i
(
ω
)
=
ω
i
i
=
1
,
…
,
4
G
(
Q
(
ω
)
/
Q
)
[
Q
(
ω
)
:
Q
]
=
|
G
(
Q
(
ω
)
/
Q
)
|
=
4
σ
i
G
(
Q
(
ω
)
/
Q
)
G
(
Q
(
ω
)
/
Q
)
≅
Z
4
ω
f
(
x
)
F
[
x
]
E
f
(
x
)
F
[
x
]
f
(
x
)
E
f
(
x
)
=
(
x
−
α
1
)
n
1
(
x
−
α
2
)
n
2
⋯
(
x
−
α
r
)
n
r
=
∏
i
=
1
r
(
x
−
α
i
)
n
i
α
i
f
(
x
)
n
i
f
(
x
)
∈
F
[
x
]
n
n
E
f
(
x
)
E
[
x
]
E
F
F
E
F
[
x
]
f
(
x
)
gcd
(
f
(
x
)
,
f
′
(
x
)
)
=
1
f
(
x
)
F
F
0
f
(
x
)
F
p
f
(
x
)
g
(
x
p
)
g
(
x
)
F
[
x
]
f
(
x
)
char
F
=
0
deg
f
′
(
x
)
<
deg
f
(
x
)
f
(
x
)
gcd
(
f
(
x
)
,
f
′
(
x
)
)
1
f
′
(
x
)
char
F
=
p
f
′
(
x
)
f
′
(
x
)
p
f
(
x
)
=
a
0
+
a
1
x
p
+
a
2
x
2
p
+
⋯
+
a
n
x
n
p
F
F
(
α
)
E
F
α
∈
E
E
=
F
(
α
)
α
Q
(
3
,
5
)
=
Q
(
3
+
5
)
Q
(
5
3
,
5
i
)
=
Q
(
5
6
i
)
E
F
α
∈
E
E
=
F
(
α
)
F
E
F
(
α
,
β
)
f
(
x
)
g
(
x
)
α
β
K
f
(
x
)
g
(
x
)
f
(
x
)
α
=
α
1
,
…
,
α
n
K
g
(
x
)
β
=
β
1
,
…
,
β
m
K
1
E
F
F
a
F
a
α
i
−
α
β
−
β
j
i
j
j
1
a
(
β
−
β
j
)
α
i
−
α
γ
=
α
+
a
β
γ
=
α
+
a
β
α
i
+
a
β
j
;
γ
−
a
β
j
α
i
i
,
j
j
1
h
(
x
)
∈
F
(
γ
)
[
x
]
h
(
x
)
=
f
(
γ
−
a
x
)
h
(
β
)
=
f
(
α
)
=
0
h
(
β
j
)
0
j
1
h
(
x
)
g
(
x
)
F
(
γ
)
[
x
]
β
F
(
γ
)
β
g
(
x
)
h
(
x
)
β
∈
F
(
γ
)
α
=
γ
−
a
β
F
(
γ
)
F
(
α
,
β
)
=
F
(
γ
)
G
(
E
/
F
)
E
F
{
σ
i
:
i
∈
I
}
F
σ
i
F
{
σ
i
}
=
{
a
∈
F
:
σ
i
(
a
)
=
a
for all
σ
i
}
F
σ
i
(
a
)
=
a
σ
i
(
b
)
=
b
σ
i
(
a
±
b
)
=
σ
i
(
a
)
±
σ
i
(
b
)
=
a
±
b
σ
i
(
a
b
)
=
σ
i
(
a
)
σ
i
(
b
)
=
a
b
a
0
σ
i
(
a
−
1
)
=
[
σ
i
(
a
)
]
−
1
=
a
−
1
σ
i
(
0
)
=
0
σ
i
(
1
)
=
1
σ
i
F
G
Aut
(
F
)
G
F
G
=
{
α
∈
F
:
σ
(
α
)
=
α
for all
σ
∈
G
}
F
F
{
σ
i
}
F
{
σ
i
}
G
Aut
(
F
)
F
G
σ
:
Q
(
3
,
5
)
→
Q
(
3
,
5
)
3
−
3
Q
(
5
)
Q
(
3
,
5
)
σ
E
F
E
G
(
E
/
F
)
=
F
G
=
G
(
E
/
F
)
F
⊂
E
G
⊂
E
E
E
G
G
(
E
/
F
)
=
G
(
E
/
E
G
)
|
G
|
=
[
E
:
E
G
]
=
[
E
:
F
]
[
E
G
:
F
]
=
1
E
G
=
F
G
E
F
=
E
G
[
E
:
F
]
≤
|
G
|
|
G
|
=
n
n
+
1
α
1
,
…
,
α
n
+
1
E
F
a
i
∈
F
a
1
α
1
+
a
2
α
2
+
⋯
+
a
n
+
1
α
n
+
1
=
0
σ
1
=
i
d
,
σ
2
,
…
,
σ
n
G
σ
1
(
α
1
)
x
1
+
σ
1
(
α
2
)
x
2
+
⋯
+
σ
1
(
α
n
+
1
)
x
n
+
1
=
0
σ
2
(
α
1
)
x
1
+
σ
2
(
α
2
)
x
2
+
⋯
+
σ
2
(
α
n
+
1
)
x
n
+
1
=
0
⋮
σ
n
(
α
1
)
x
1
+
σ
n
(
α
2
)
x
2
+
⋯
+
σ
n
(
α
n
+
1
)
x
n
+
1
=
0
x
i
=
a
i
i
=
1
,
2
,
…
,
n
+
1
σ
1
a
1
α
1
+
a
2
α
2
+
⋯
+
a
n
+
1
α
n
+
1
=
0
a
i
E
F
a
i
E
F
α
i
a
1
a
1
=
1
α
2
,
…
,
α
n
+
1
a
2
E
F
F
E
G
σ
i
G
σ
i
(
a
2
)
a
2
σ
i
G
x
1
=
σ
i
(
a
1
)
=
1
x
2
=
σ
i
(
a
2
)
…
x
n
+
1
=
σ
i
(
a
n
+
1
)
x
1
=
1
−
1
=
0
x
2
=
a
2
−
σ
i
(
a
2
)
⋮
x
n
+
1
=
a
n
+
1
−
σ
i
(
a
n
+
1
)
σ
i
(
a
2
)
a
2
a
1
,
…
,
a
n
+
1
∈
F
E
F
F
[
x
]
E
E
E
F
F
[
x
]
E
E
[
x
]
E
F
E
F
E
F
F
=
E
G
G
E
⇒
E
F
α
E
E
=
F
(
α
)
f
(
x
)
α
F
E
f
(
x
)
F
E
f
(
x
)
⇒
E
F
E
G
(
E
/
F
)
=
F
|
G
(
E
/
F
)
|
=
[
E
:
F
]
⇒
F
=
E
G
G
E
[
E
:
F
]
≤
|
G
|
E
F
E
F
f
(
x
)
∈
F
[
x
]
α
E
f
(
x
)
E
[
x
]
G
f
(
x
)
E
G
α
α
1
=
α
,
α
2
,
…
,
α
n
E
g
(
x
)
=
∏
i
=
1
n
(
x
−
α
i
)
g
(
x
)
F
g
(
α
)
=
0
σ
G
g
(
x
)
σ
g
(
x
)
g
(
x
)
g
(
x
)
F
deg
g
(
x
)
≤
deg
f
(
x
)
f
(
x
)
α
f
(
x
)
=
g
(
x
)
K
F
F
=
K
G
G
K
G
=
G
(
K
/
F
)
F
=
K
G
G
G
(
K
/
F
)
[
K
:
F
]
≤
|
G
|
≤
|
G
(
K
/
F
)
|
=
[
K
:
F
]
G
=
G
(
K
/
F
)
Q
(
3
,
5
)
Q
Q
G
(
Q
(
3
,
5
)
/
Q
)
G
(
Q
(
3
,
5
)
/
Q
)
F
E
F
G
(
E
/
F
)
K
↦
G
(
E
/
K
)
K
E
F
G
(
E
/
F
)
F
⊂
K
⊂
E
[
E
:
K
]
=
|
G
(
E
/
K
)
|
and
[
K
:
F
]
=
[
G
(
E
/
F
)
:
G
(
E
/
K
)
]
F
⊂
K
⊂
L
⊂
E
{
i
d
}
⊂
G
(
E
/
L
)
⊂
G
(
E
/
K
)
⊂
G
(
E
/
F
)
K
F
G
(
E
/
K
)
G
(
E
/
F
)
G
(
K
/
F
)
≅
G
(
E
/
F
)
/
G
(
E
/
K
)
G
(
E
/
K
)
=
G
(
E
/
L
)
=
G
K
L
G
K
=
L
K
↦
G
(
E
/
K
)
G
G
(
E
/
F
)
K
G
F
⊂
K
⊂
E
E
K
G
(
E
/
K
)
=
G
K
↦
G
(
E
/
K
)
|
G
(
E
/
K
)
|
=
[
E
:
K
]
|
G
(
E
/
F
)
|
=
[
G
(
E
/
F
)
:
G
(
E
/
K
)
]
⋅
|
G
(
E
/
K
)
|
=
[
E
:
F
]
=
[
E
:
K
]
[
K
:
F
]
[
K
:
F
]
=
[
G
(
E
/
F
)
:
G
(
E
/
K
)
]
K
F
σ
G
(
E
/
F
)
τ
G
(
E
/
K
)
σ
−
1
τ
σ
G
(
E
/
K
)
σ
−
1
τ
σ
(
α
)
=
α
α
∈
K
f
(
x
)
α
F
σ
(
α
)
f
(
x
)
K
K
F
τ
(
σ
(
α
)
)
=
σ
(
α
)
σ
−
1
τ
σ
(
α
)
=
α
G
(
E
/
K
)
G
(
E
/
F
)
F
=
K
G
(
K
/
F
)
τ
∈
G
(
E
/
K
)
σ
∈
G
(
E
/
F
)
τ
¯
∈
G
(
E
/
K
)
τ
σ
=
σ
τ
¯
α
∈
K
τ
(
σ
(
α
)
)
=
σ
(
τ
¯
(
α
)
)
=
σ
(
α
)
;
σ
(
α
)
G
(
E
/
K
)
σ
¯
σ
K
σ
¯
K
F
σ
(
α
)
∈
K
α
∈
K
σ
¯
∈
G
(
K
/
F
)
G
(
K
/
F
)
F
β
K
G
(
K
/
F
)
σ
¯
(
β
)
=
β
σ
∈
G
(
E
/
F
)
β
F
G
(
E
/
F
)
K
F
G
(
K
/
F
)
≅
G
(
E
/
F
)
/
G
(
E
/
K
)
σ
∈
G
(
E
/
F
)
σ
K
K
σ
K
K
σ
K
∈
G
(
K
/
F
)
ϕ
:
G
(
E
/
F
)
→
G
(
K
/
F
)
σ
↦
σ
K
ϕ
(
σ
τ
)
=
(
σ
τ
)
K
=
σ
K
τ
K
=
ϕ
(
σ
)
ϕ
(
τ
)
ϕ
G
(
E
/
K
)
|
G
(
E
/
F
)
|
/
|
G
(
E
/
K
)
|
=
[
K
:
F
]
=
|
G
(
K
/
F
)
|
ϕ
G
(
K
/
F
)
ϕ
G
(
K
/
F
)
≅
G
(
E
/
F
)
/
G
(
E
/
K
)
G
(
E
/
F
)
E
E
→
{
id
}
↑
↓
L
→
G
(
E
/
L
)
↑
↓
K
→
G
(
E
/
K
)
↑
↓
F
→
G
(
E
/
F
)
f
(
x
)
=
x
4
−
2
Q
x
4
−
2
f
(
x
)
Q
(
2
4
,
i
)
f
(
x
)
(
x
2
+
2
)
(
x
2
−
2
)
f
(
x
)
±
2
4
±
2
4
i
2
4
Q
i
x
2
+
1
Q
(
2
4
)
f
(
x
)
Q
(
2
4
)
(
i
)
=
Q
(
2
4
,
i
)
[
Q
(
2
4
)
:
Q
]
=
4
i
Q
(
2
4
)
[
Q
(
2
4
,
i
)
:
Q
(
2
4
)
]
=
2
[
Q
(
2
4
,
i
)
:
Q
]
=
8
{
1
,
2
4
,
(
2
4
)
2
,
(
2
4
)
3
,
i
,
i
2
4
,
i
(
2
4
)
2
,
i
(
2
4
)
3
}
Q
(
2
4
,
i
)
Q
Q
Q
(
2
4
,
i
)
G
f
(
x
)
8
σ
σ
(
2
4
)
=
i
2
4
σ
(
i
)
=
i
τ
τ
(
i
)
=
−
i
G
4
2
G
{
i
d
,
σ
,
σ
2
,
σ
3
,
τ
,
σ
τ
,
σ
2
τ
,
σ
3
τ
}
τ
2
=
i
d
σ
4
=
i
d
τ
σ
τ
=
σ
−
1
G
D
4
G
x
4
−
2
f
(
x
)
f
(
x
)
n
a
x
2
+
b
x
+
c
=
0
x
=
−
b
±
b
2
−
4
a
c
2
a
n
E
F
F
=
F
0
⊂
F
1
⊂
F
2
⊂
⋯
⊂
F
r
=
E
i
=
1
,
2
,
…
,
r
F
i
=
F
i
−
1
(
α
i
)
α
i
n
i
∈
F
i
−
1
n
i
f
(
x
)
F
K
f
(
x
)
F
F
f
(
x
)
f
(
x
)
x
n
−
a
x
n
−
1
n
x
n
−
1
n
n
x
n
−
1
Q
1
,
ω
,
ω
2
,
…
,
ω
n
−
1
ω
=
cos
(
2
π
n
)
+
i
sin
(
2
π
n
)
x
n
−
1
Q
Q
(
ω
)
G
G
=
H
n
⊃
H
n
−
1
⊃
⋯
⊃
H
1
⊃
H
0
=
{
e
}
H
i
H
i
+
1
G
{
H
i
}
H
i
+
1
/
H
i
{
i
d
}
⊂
A
3
⊂
S
3
S
3
S
5
F
E
x
n
−
a
F
a
∈
F
G
(
E
/
F
)
x
n
−
a
a
n
,
ω
a
n
,
…
,
ω
n
−
1
a
n
ω
n
F
n
ζ
x
n
−
a
x
n
−
a
ζ
,
ω
ζ
,
…
,
ω
n
−
1
ζ
E
=
F
(
ζ
)
G
(
E
/
F
)
x
n
−
a
G
(
E
/
F
)
σ
τ
G
(
E
/
F
)
σ
(
ζ
)
=
ω
i
ζ
τ
(
ζ
)
=
ω
j
ζ
F
σ
τ
(
ζ
)
=
σ
(
ω
j
ζ
)
=
ω
j
σ
(
ζ
)
=
ω
i
+
j
ζ
=
ω
i
τ
(
ζ
)
=
τ
(
ω
i
ζ
)
=
τ
σ
(
ζ
)
σ
τ
=
τ
σ
G
(
E
/
F
)
G
(
E
/
F
)
F
n
ω
n
α
x
n
−
a
α
ω
α
x
n
−
a
ω
=
(
ω
α
)
/
α
E
K
=
F
(
ω
)
F
⊂
K
⊂
E
K
x
n
−
1
K
F
σ
G
(
F
(
ω
)
/
F
)
σ
(
ω
)
σ
(
ω
)
=
ω
i
i
x
n
−
1
ω
τ
(
ω
)
=
ω
j
G
(
F
(
ω
)
/
F
)
σ
τ
(
ω
)
=
σ
(
ω
j
)
=
[
σ
(
ω
)
]
j
=
ω
i
j
=
[
τ
(
ω
)
]
i
=
τ
(
ω
i
)
=
τ
σ
(
ω
)
G
(
F
(
ω
)
/
F
)
{
i
d
}
⊂
G
(
E
/
F
(
ω
)
)
⊂
G
(
E
/
F
)
G
(
E
/
F
(
ω
)
)
G
(
E
/
F
)
/
G
(
E
/
F
(
ω
)
)
≅
G
(
F
(
ω
)
/
F
)
G
(
E
/
F
)
F
F
=
F
0
⊂
F
1
⊂
F
2
⊂
⋯
⊂
F
r
=
E
F
F
=
K
0
⊂
K
1
⊂
K
2
⊂
⋯
⊂
K
r
=
K
K
E
K
i
K
i
−
1
E
F
F
=
F
0
⊂
F
1
⊂
F
2
⊂
⋯
⊂
F
r
=
E
i
=
1
,
2
,
…
,
r
F
i
=
F
i
−
1
(
α
i
)
α
i
n
i
∈
F
i
−
1
n
i
F
F
=
K
0
⊂
K
1
⊂
K
2
⊂
⋯
⊂
K
r
=
K
K
⊇
E
K
1
x
n
1
−
α
1
n
1
α
1
,
α
1
ω
,
α
1
ω
2
,
…
,
α
1
ω
n
1
−
1
ω
n
1
F
n
1
K
1
=
F
(
α
!
)
F
n
1
β
x
n
1
−
α
1
n
1
x
n
1
−
α
1
n
1
β
,
ω
β
,
…
,
ω
n
1
−
1
ω
n
1
K
1
=
F
(
ω
β
)
K
1
F
F
1
F
=
K
0
⊂
K
1
⊂
K
2
⊂
⋯
⊂
K
r
=
K
K
i
K
i
−
1
K
i
⊇
F
i
i
=
1
,
2
,
…
,
r
f
(
x
)
F
[
x
]
char
F
=
0
f
(
x
)
f
(
x
)
F
f
(
x
)
E
F
F
=
F
0
⊂
F
1
⊂
⋯
⊂
F
n
=
E
E
f
(
x
)
F
i
F
i
−
1
G
(
E
/
F
i
)
G
(
E
/
F
i
−
1
)
G
(
E
/
F
)
{
i
d
}
⊂
G
(
E
/
F
n
−
1
)
⊂
⋯
⊂
G
(
E
/
F
1
)
⊂
G
(
E
/
F
)
G
(
E
/
F
i
−
1
)
/
G
(
E
/
F
i
)
≅
G
(
F
i
/
F
i
−
1
)
G
(
F
i
/
F
i
−
1
)
G
(
E
/
F
)
S
5
p
S
p
p
S
p
G
S
p
σ
τ
p
σ
=
(
1
2
)
τ
p
τ
n
p
1
≤
n
<
p
μ
=
τ
n
=
(
1
2
i
3
…
i
p
)
n
1
≤
n
<
p
(
1
2
)
(
12
i
3
…
i
p
)
=
(
2
i
3
…
i
p
)
(
2
i
3
…
i
p
)
k
(
12
)
(
2
i
3
…
i
p
)
−
k
=
(
1
i
k
)
(
1
n
)
1
≤
n
<
p
S
p
(
1
j
)
(
1
i
)
(
1
j
)
=
(
i
j
)
S
p
f
(
x
)
=
x
5
−
6
x
3
−
27
x
−
3
f
(
x
)
=
x
5
−
6
x
3
−
27
x
−
3
∈
Q
[
x
]
f
(
x
)
Q
S
5
f
(
x
)
f
(
x
)
f
′
(
x
)
=
5
x
4
−
18
x
2
−
27
f
′
(
x
)
=
0
f
′
(
x
)
x
=
±
6
6
+
9
5
f
(
x
)
f
(
x
)
−
3
−
2
−
2
0
0
4
f
(
x
)
f
(
x
)
K
f
(
x
)
f
(
x
)
K
K
Q
f
(
x
)
G
(
K
/
Q
)
S
5
f
σ
∈
G
(
K
/
Q
)
σ
(
a
)
=
b
a
b
f
(
x
)
C
a
+
b
i
↦
a
−
b
i
G
(
K
/
Q
)
α
f
(
x
)
[
Q
(
α
)
:
Q
]
=
5
Q
(
α
)
K
[
K
:
Q
]
[
K
:
Q
]
=
|
G
(
K
/
Q
)
|
G
(
K
/
Q
)
⊂
S
5
G
(
K
/
Q
)
5
S
5
5
G
(
K
/
Q
)
S
5
S
5
f
(
x
)
C
[
x
]
C
E
α
∈
E
E
=
C
(
α
)
α
f
(
x
)
C
[
x
]
L
f
(
x
)
C
E
L
C
L
C
L
f
(
x
)
(
x
2
+
1
)
R
L
R
K
G
G
(
L
/
R
)
L
⊃
K
⊃
R
|
G
(
L
/
K
)
|
=
[
L
:
K
]
[
L
:
R
]
=
[
L
:
K
]
[
K
:
R
]
[
K
:
R
]
K
=
R
(
β
)
β
f
(
x
)
K
=
R
G
(
L
/
R
)
G
(
L
/
C
)
2
L
C
|
G
(
L
/
C
)
|
≥
2
G
G
(
L
/
C
)
E
G
[
E
:
C
]
=
2
γ
∈
E
x
2
+
b
x
+
c
C
[
x
]
(
−
b
±
b
2
−
4
c
)
/
2
C
b
2
−
4
c
C
L
=
C
Q
G
(
Q
(
30
)
/
Q
)
G
(
Q
(
5
4
)
/
Q
)
G
(
Q
(
2
,
3
,
5
)
/
Q
)
G
(
Q
(
2
,
2
3
,
i
)
/
Q
)
G
(
Q
(
6
,
i
)
/
Q
)
Z
2
Z
2
×
Z
2
×
Z
2
x
3
+
2
x
2
−
x
−
2
Q
x
4
+
2
x
2
+
1
Q
x
4
+
x
2
+
1
Z
3
x
3
+
x
2
+
1
Z
2
Q
x
3
+
2
x
2
−
x
−
2
=
(
x
−
1
)
(
x
+
1
)
(
x
+
2
)
Z
3
x
4
+
x
2
+
1
=
(
x
+
1
)
2
(
x
+
2
)
2
GF
(
729
)
GF
(
9
)
[
GF
(
729
)
:
GF
(
9
)
]
=
[
GF
(
729
)
:
GF
(
3
)
]
/
[
GF
(
9
)
:
GF
(
3
)
]
=
6
/
2
=
3
G
(
GF
(
729
)
/
GF
(
9
)
)
≅
Z
3
G
(
GF
(
729
)
/
GF
(
9
)
)
σ
σ
3
6
(
α
)
=
α
3
6
=
α
729
α
∈
GF
(
729
)
Q
[
x
]
x
5
−
12
x
2
+
2
x
5
−
4
x
4
+
2
x
+
2
x
3
−
5
x
4
−
x
2
−
6
x
5
+
1
(
x
2
−
2
)
(
x
2
+
2
)
x
8
−
1
x
8
+
1
x
4
−
3
x
2
−
10
S
5
S
3
Q
[
x
]
x
4
−
1
x
4
−
8
x
2
+
15
x
4
−
2
x
2
−
15
x
3
−
2
Q
(
i
)
Z
2
S
3
Z
3
E
F
[
x
]
[
E
:
F
]
6
3
G
(
E
/
F
)
S
3
3
Z
3
S
3
F
⊂
K
⊂
E
E
F
E
K
G
n
|
G
|
n
!
G
S
n
F
⊂
E
f
(
x
)
F
f
(
x
)
E
f
(
x
)
Q
[
x
]
7
p
f
(
x
)
∈
Q
[
x
]
p
S
p
p
p
≥
5
p
p
Z
p
(
t
)
Z
p
f
(
x
)
=
x
p
−
t
Z
p
(
t
)
[
x
]
f
(
x
)
E
F
K
L
σ
∈
G
(
E
/
F
)
σ
(
K
)
=
L
K
L
K
L
G
(
E
/
K
)
G
(
E
/
L
)
G
(
E
/
F
)
σ
∈
Aut
(
R
)
a
σ
(
a
)
>
0
K
x
3
+
x
2
+
1
∈
Z
2
[
x
]
K
F
char
(
F
)
2
f
(
x
)
=
a
x
2
+
b
x
+
c
F
(
α
)
α
=
b
2
−
4
a
c
K
F
E
F
K
[
E
:
F
]
=
2
E
F
[
x
]
Φ
p
(
x
)
=
x
p
−
1
x
−
1
=
x
p
−
1
+
x
p
−
2
+
⋯
+
x
+
1
Q
p
ω
Φ
p
(
x
)
Q
(
ω
)
ω
,
ω
2
,
…
,
ω
p
−
1
Φ
p
(
x
)
Φ
p
(
x
)
G
(
Q
(
ω
)
/
Q
)
p
−
1
G
(
Q
(
ω
)
/
Q
)
Q
ω
,
ω
2
,
…
,
ω
p
−
1
ω
1
ω
i
Φ
p
Φ
p
(
ω
i
)
ω
ω
,
ω
2
,
…
,
ω
p
−
1
ϕ
i
:
Q
(
ω
)
→
Q
(
ω
i
)
ϕ
i
(
a
0
+
a
1
ω
+
⋯
+
a
p
−
2
ω
p
−
2
)
=
a
0
+
a
1
ω
i
+
⋯
+
c
p
−
2
(
ω
i
)
p
−
2
a
i
∈
Q
ϕ
i
ϕ
2
G
(
Q
(
ω
)
/
Q
)
{
ω
,
ω
2
,
…
,
ω
p
−
1
}
Q
(
ω
)
Q
ω
,
ω
2
,
…
,
ω
p
−
1
G
(
Q
(
ω
)
/
Q
)
F
E
F
G
(
E
/
F
)
F
⊂
K
⊂
L
⊂
E
{
i
d
}
⊂
G
(
E
/
L
)
⊂
G
(
E
/
K
)
⊂
G
(
E
/
F
)
F
f
(
x
)
∈
F
[
x
]
n
E
f
(
x
)
α
1
,
…
,
α
n
f
(
x
)
E
Δ
=
∏
i
<
j
(
α
i
−
α
j
)
f
(
x
)
Δ
2
f
(
x
)
=
x
2
+
b
x
+
c
Δ
2
=
b
2
−
4
c
f
(
x
)
=
x
3
+
p
x
+
q
Δ
2
=
−
4
p
3
−
27
q
2
Δ
2
F
σ
∈
G
(
E
/
F
)
f
(
x
)
σ
(
Δ
)
=
−
Δ
σ
∈
G
(
E
/
F
)
f
(
x
)
σ
(
Δ
)
=
Δ
G
(
E
/
F
)
A
n
Δ
∈
F
x
3
+
2
x
−
4
x
3
+
x
−
3
p
(
x
)
=
x
4
−
2
8
8
8
8
p
(
x
)
=
x
4
−
2
2
1
4
=
2
4
2
1
4
i
=
2
4
i
S
4
8
8
τ
(
x
)
=
τ
(
∑
i
=
0
7
q
i
c
i
)
q
i
∈
Q
=
∑
i
=
0
7
τ
(
q
i
)
τ
(
c
i
)
τ
is a field automorphism
=
∑
i
=
0
7
q
i
τ
(
c
i
)
rationals are fixed
τ
τ
(
c
)
τ
(
c
k
)
=
(
τ
(
c
)
)
k
2
1
2
4
i
2
2
4
i
Q
(
2
4
i
)
Q
(
2
4
i
)
2
4
i
−
2
4
=
(
1
−
i
)
2
4
Q
(
2
4
i
−
2
4
)
=
Q
(
(
1
−
i
)
2
4
)
x
4
+
8
(
1
−
i
)
2
4
x
4
+
8
2
2
2
Z
2
×
Z
2
2
Q
(
2
)
−
14
2
i
i
Q
(
i
)
4
2
Q
(
2
4
)
x
4
−
2
Q
(
2
4
)
H
=
⟨
(
1
,
4
)
⟩
Q
(
2
4
)
x
4
−
2
2
4
p
(
x
)
=
x
3
−
6
x
2
+
12
x
−
10
p
(
x
)
x
5
−
x
−
1
S
5
S
5
p
(
x
)
=
x
4
+
x
+
1
p
(
x
)
p
(
x
)
Q
p
(
x
)
Q
x
5
−
x
−
1
3
6
x
6
+
x
2
+
2
x
+
1
p
(
x
)
=
x
7
−
7
x
+
3
y
2
=
x
(
81
x
5
+
396
x
4
+
738
x
3
+
660
x
2
+
269
x
+
48
)
p
(
x
)
P
S
L
(
2
,
7
)
S
L
(
2
,
7
)
2
×
2
Z
7
P
S
L
(
2
,
7
)
{
I
2
,
−
I
2
}
A
5
7
6
168